Introduction

The theory of slice regular functions of one variable in a real alternative \(^*\)-algebra is now well developed. It was introduced firstly for functions of one quaternionic variable by Gentili and Struppa in [5, 6] and then extended to octonions in [7] and to Clifford algebras in [2]. In [8, 9], a new approach to slice regularity, based on the concept of stem function, allowed to extend the theory to any real alternative \(^*\)-algebra A of finite dimension.

The original definition [5, 6] of slice regularity for a quaternion-valued function f, defined on an open domain \(\Omega \) of the algebra \({\mathbb {H}}\) of quaternions, requires that, for every imaginary unit \(J\in {\mathbb {H}}\), the restriction of f to the complex line generated by J is holomorphic with respect to the complex structure defined by left multiplication by J. The approach taken in [8, 9] allows to embed the class of slice regular functions on an axially symmetric domain into a larger class, that of slice functions, on which no holomorphicity condition is assumed. We refer to the monograph [4] for a survey of slice analysis in one quaternionic variable and to the papers [13,14,15] for a recent account of the theory on real alternative \(^*\)-algebras.

In the present paper we propose a generalization of slice analysis to several variables in a real alternative \(^*\)-algebra A. Our function theory includes, in particular, the class of polynomials in several (ordered) variables with right coefficients in A. Our approach is based on the concept of stem functions of several variables and on the introduction of a family of commuting complex structures on the real vector space \(\mathbb {R}^{2^n}\). For \(A={\mathbb {H}}\), several variables have been investigated also by Colombo, Sabadini and Struppa [1]. Their approach via stem functions is similar to ours, but the definition of regularity is different, as we will see in Sect. 3. For \(A={\mathbb {O}}\), the algebra of octonions, a slice functions theory of several variables has been proposed by Ren and Yang [21]. The major difference with our theory is that the authors define slice functions on a class of non-open subsets of the space \({\mathbb {O}}^n\), where the octonionic variables associate and commute. In [21] the authors state as challenging the possibility to establish the theory on open subsets of \({\mathbb {O}}^n\). For \(A=\mathbb {C}\), as one may expect, the slice function theory in n variables reduces to the classic theory of several complex variables on domains D of \(\mathbb {C}^n\), with the unique restriction that D must be assumed invariant with respect to complex conjugation in every variable \(z_1,\ldots ,z_n\).

Some of the results proved here were presented in [10] for the case \(A=\mathbb {R}_m\), the real Clifford algebra of signature (0, m), and in [11] for the general case of real alternative \(^*\)-algebras. See also [23] for a review of quaternionic and Clifford Analysis in several variables.

We describe the structure of the paper. In the Introduction we give some preliminaries and recall the main definitions of the one variable slice function theory. Then we present without proofs the principal results in two quaternionic variables, where the exposition is simpler but sufficiently representative of the theory, at least for the associative case. In Sect. 2, we introduce the stem functions of several variables in A and define the induced slice functions. We prove a representation formula and the identity principle. We generalize to n variables the concepts of spherical value and spherical derivatives and give the relation between sliceness in n variables and sliceness in one variable. We then prove the smoothness properties of slice functions. We study also the multiplicative structures on slice functions induced by pointwise products of stem functions, and we investigate some special real subalgebras of slice functions. Section 3 is dedicated to slice regularity. After giving the definition of a family of commuting complex structures on \(A\otimes \mathbb {R}^{2^n}\), the concept of slice regular function of several variables is introduced. All polynomials (with ordered monomials) turn out to be slice regular functions. We study the real dimension of the zero set of polynomials in the quaternionic and octonionic cases and give some results about the zero set of polynomials with Clifford coefficients. In particular, we prove that these zero sets are nonempty for nonconstant polynomials. We show that slice regularity in several variables has an interpretation, by means of the spherical value and spherical derivatives, in terms of slice regularity in one variable. We investigate Leibniz’s rule and we prove the stability of slice regularity under the so-called slice tensor product of slice functions. We show the relation between slice regularity and expansions in (ordered) power series on products of open balls in A centered in the origin. Finally, we define a slice Cauchy kernel associated to any given slice regular function, and obtain a Cauchy integral formula. In the associative case, we are able to define a universal slice Cauchy kernel, and express it in terms of pointwise operations in A.

Preliminaries

Let A be a real algebra with unity \(1\ne 0\). Assume that A is alternative, i.e. \(x^2y=x(xy)\) and \((yx)x=yx^2\) for all \(x,y\in A\). A theorem of E. Artin asserts that the subalgebra generated by any two elements of A is associative. The real multiples of 1 in A are identified with the field \(\mathbb {R}\) of real numbers. Assume that A is a *-algebra, i.e., it is equipped with a real linear anti-involution \(A\rightarrow A\), \(x\mapsto x^c\), such that \((xy)^c=y^cx^c\) for all \(x,y\in A\) and \(x^c=x\) for x real. Let \(t(x):=x+x^c\in A\) be the trace of x and \(n(x):=xx^c\in A\) the (squared) norm of x. Let

$$\begin{aligned} {\mathbb {S}}_A:=\{J\in A : t(x)=0,\ n(x)=1\} \end{aligned}$$

be the ‘sphere’ of the imaginary units of A compatible with the *-algebra structure of A. Assuming \({\mathbb {S}}_A\ne \emptyset \), one can consider the quadratic cone of A, defined as the subset of A

$$\begin{aligned} Q_A:=\bigcup _{J\in {\mathbb {S}}_A}\mathbb {C}_J, \end{aligned}$$

where \(\mathbb {C}_J=\langle 1,J\rangle \) is the complex ‘slice’ of A generated by 1, J as a vector subspace or, equivalently, by J as a subalgebra. It holds \(\mathbb {C}_J\cap \mathbb {C}_K=\mathbb {R}\) for each \(J,K\in {\mathbb {S}}_A\) with \(J\ne \pm K\). The quadratic cone is a real cone invariant w.r.t. translations along the real axis. Observe that t and n are real-valued on \(Q_A\) and that \(Q_A=A\) if and only if A is isomorphic as a real \(^*\)-algebra to one of the division algebras \(\mathbb {C},{\mathbb {H}},{\mathbb {O}}\) with the standard conjugations (see [9,  Proposition 1]). Moreover, it holds

$$\begin{aligned} Q_A=\mathbb {R}\cup \{x\in A\setminus \mathbb {R}: t(x)\in \mathbb {R},n(x)\in \mathbb {R},4n(x)>t(x)^2\}. \end{aligned}$$

Each element x of \(Q_A\) can be written as \(x={\text {Re}}(x)+{\text {Im}}(x)\), with \({\text {Re}}(x)=\frac{x+x^c}{2}\), \({\text {Im}}(x)=\frac{x-x^c}{2}=\beta J\), where \(\beta =\sqrt{n({\text {Im}}(x))}\ge 0\) and \(J\in {\mathbb {S}}_A\). The choice of \(\beta \ge 0\) and \(J\in {\mathbb {S}}_A\) is unique if \(x\not \in \mathbb {R}\).

We refer to [9,  §2] and [13,  §1] for more details and examples about real alternative \(^*\)-algebras and their quadratic cones.

The one variable slice function theory

The slice functions on A are the functions which are compatible with the slice character of the quadratic cone. More precisely, let D be a subset of \(\mathbb {C}\) that is invariant w.r.t. complex conjugation. Let \(A\otimes _{\mathbb {R}}\mathbb {C}\) be the complexified algebra, whose elements w are of the form \(w=a+ib\) with \(a,b\in A\) and \(i^2=-1\). In \(A\otimes _{\mathbb {R}}\mathbb {C}\) we consider the complex conjugation mapping \(w=a+ib\) to \({{\overline{w}}}=a-ib\) for all \(a,b\in A\). If a function \(F: D \rightarrow A\otimes _{\mathbb {R}}\mathbb {C}\) satisfies \(F({{\overline{z}}})=\overline{F(z)}\) for every \(z\in D\), then F is called a stem function on D. For every \(J\in {\mathbb {S}}_A\), we define the real \(^*\)-algebra isomorphism \(\phi _J:\mathbb {C}\rightarrow \mathbb {C}_J\) by setting

$$\begin{aligned} \phi _J(\alpha +i\beta ):=\alpha +J\beta \,\text { for all }\alpha ,\beta \in \mathbb {R}. \end{aligned}$$
(1)

Let \(\Omega _D\) be the circular subset of the quadratic cone defined by

$$\begin{aligned} \Omega _D=\bigcup _{J\in {\mathbb {S}}_A}\phi _J(D)=\{\alpha +J\beta \in A : \alpha ,\beta \in \mathbb {R}, \alpha +i\beta \in D,J\in {\mathbb {S}}_A\}. \end{aligned}$$

The stem function \(F=F_1+iF_2:D \rightarrow A\otimes _{\mathbb {R}}\mathbb {C}\) induces the (left) slice function \(f=\mathcal {I}(F):\Omega _D \rightarrow A\) in the following way: if \(x=\alpha +J\beta =\phi _J(z)\in \Omega _D\cap \mathbb {C}_J\), then

$$\begin{aligned} f(x)=F_1(z)+JF_2(z), \end{aligned}$$

where \(z=\alpha +i\beta \).

Suppose that D is open. Left multiplication by i defines a complex structure on \(A\otimes _{\mathbb {R}}\mathbb {C}\). The slice function \(f=\mathcal {I}(F):\Omega _D \rightarrow A\) is called (left) slice regular if F is holomorphic. For example, polynomial functions \(f(x)=\sum _{j=0}^d x^ja_j\) with right coefficients belonging to A are slice regular on the quadratic cone.

To any slice function \(f=\mathcal {I}(F):\Omega _D \rightarrow A\), one can associate the function \({f}_s^\circ :\Omega _D \rightarrow A\), called spherical value of f, and the function \(f'_s:\Omega _D \setminus \mathbb {R}\rightarrow A\), called spherical derivative of f, defined as

$$\begin{aligned} {f}_s^\circ (x):=\frac{1}{2}(f(x)+f(x^c)) \quad \text {and} \quad f'_s(x):=\frac{1}{2}{\text {Im}}(x)^{-1}(f(x)-f(x^c)). \end{aligned}$$

If \(x=\alpha +\beta J\in \Omega _D\) and \(z=\alpha +i\beta \in D\), then \({f}_s^\circ (x)=F_1(z)\) and \(f'_s(x)=\beta ^{-1} F_2(z)\). Therefore \({f}_s^\circ \) and \(f'_s\) are slice functions, constant on every set \({\mathbb {S}}_x:=\alpha +\beta \,{\mathbb {S}}_A\). They are slice regular only if f is locally constant. Moreover, the formula

$$\begin{aligned} f(x)={f}_s^\circ (x)+{\text {Im}}(x)f'_s(x) \end{aligned}$$

holds for all \(x\in \Omega _D\setminus \mathbb {R}\). As we will see later, the concepts of spherical value and spherical derivative in one variable will have a central role to get a characterization of slice regularity in several variables in terms of separate one variable regularity.

We refer the reader to [9,  §§3,4] for more properties of slice functions and slice regularity in one variable.

Slice regular functions on \(\mathbb {H}^2\)

Before presenting the full theory in the general case of n variables in a real alternative \(^*\)-algebra A, in this subsection we summarize the main results in the simpler case of two quaternionic variables. We refer to the following sections for full proofs.

Slice functions on \(\mathbb {H}^2\)

Let D be a non-empty subset of \({\mathbb {C}}^2\), invariant w.r.t. complex conjugation in each variable \(z_1,z_2\). Let \(\Omega _D\) be the circular open subset of \({\mathbb {H}}^2\) associated to \(D\subset \mathbb {C}^2\), defined as

$$\begin{aligned} \Omega _D:=\{(\alpha _1+J_1\beta _1,\alpha _2+J_2\beta _2)\in {\mathbb {H}}^2 : J_1,J_2\in {\mathbb {S}}_{\mathbb {H}}, (\alpha _1+i\beta _1,\alpha _2+i\beta _2)\in D\}. \end{aligned}$$

Let \(\{e_\emptyset ,e_1,e_2,e_{12}\}\) denote a fixed (real vector) basis of \(\mathbb {R}^4\). If \(\mathcal {P}(2)\) denotes the set of all subsets of \(\{1,2\}\), we can write any element x of the real vector space \({\mathbb {H}}\otimes \mathbb {R}^4\) as \(x=\sum _{K\in \mathcal {P}(2)}e_Ka_K\), where each \(a_K\) belongs to \({\mathbb {H}}\), and \(e_{\{1\}}=e_1\), \(e_{\{2\}}=e_2\) and \(e_{\{1,2\}}=e_{12}\).

Definition 1.1

A function \(F:D \rightarrow {\mathbb {H}}\otimes \mathbb {R}^4\), with \(F=e_\emptyset F_\emptyset +e_1F_1+e_2F_2+e_{12}F_{12}\) and \(F_K:D\rightarrow {\mathbb {H}}\) for each \(K\in \mathcal {P}(2)\), is called a stem function if the components \(F_\emptyset , F_1, F_2, F_{12}\) are, respectively, even-even, odd-even, even-odd, odd-odd w.r.t. the pair \((\beta _1,\beta _2)\), where \(z_1=\alpha _1+i\beta _1\) and \(z_2=\alpha _2+i\beta _2\) with \(\alpha _1,\alpha _2,\beta _1,\beta _2\in \mathbb {R}\). The (left) slice function \(f=\mathcal {I}(F):\Omega _D \rightarrow {\mathbb {H}}\) induced by F is the function obtained by setting, for each \(x=(x_1,x_2)=(\alpha _1+J_1\beta _1,\alpha _2+J_2\beta _2)\),

$$\begin{aligned} \textstyle f(x):=F_\emptyset (z_1,z_2)+J_1F_1(z_1,z_2)+J_2F_2(z_1,z_2)+J_1J_2F_{12}(z_1,z_2) \end{aligned}$$

where \((z_1,z_2)=(\alpha _1+i\beta _1,\alpha _2+i\beta _2)\in D\). \(\square \)

Representation formula on \({\mathbb {H}}^2\)

The values of a slice function can be recovered by its values on a four-dimensional slice of \(\Omega _D\). Let \(a^c\) denote the conjugate of a quaternion \(a\in {\mathbb {H}}\).

Proposition 1.2

Let \(f:\Omega _D\rightarrow {\mathbb {H}}\) be a slice function and let \(y=(y_1,y_2)=(\alpha _1+I_1\beta _1,\alpha _2+I_2\beta _2)\in \Omega _D\). Then for every \(x=(x_1,x_2)=(\alpha _1+J_1\beta _1,\alpha _2+J_2\beta _2)\in \Omega _D\) it holds:

$$\begin{aligned} f(x)=&\,\frac{1}{4}\big ( f(y_1,y_2)+f(y_1^c,y_2)+f(y_1,y_2^c)+f(y_1^c,y_2^c)\\&+J_1I_1\left( -f(y_1,y_2)+ f(y_1^c,y_2)- f(y_1,y_2^c)+ f(y_1^c,y_2^c)\right) \\&+J_2I_2\left( -f(y_1,y_2)- f(y_1^c,y_2)+ f(y_1,y_2^c)+ f(y_1^c,y_2^c)\right) \\&+J_1J_2I_2I_1\left( f(y_1,y_2)- f(y_1^c,y_2)- f(y_1,y_2^c)+ f(y_1^c,y_2^c)\right) \big ). \end{aligned}$$

Corollary 1.3

(Identity principle) Let \(f,g:\Omega _D \rightarrow A\) be slice functions and let \(I_1,I_2 \in \mathbb {S}_{\mathbb {H}}\) such that \(f=g\) on \(\Omega _D \cap (\mathbb {C}_{I_1}\times \mathbb {C}_{I_2})\). Then \(f=g\) on the whole \(\Omega _D\).

Smoothness

Suppose that D is open in \(\mathbb {C}^2\).

Proposition 1.4

For every slice function \(f=\mathcal {I}(F):\Omega _D\rightarrow {\mathbb {H}}\), it holds:

  • \((\mathrm {i})\) If F is continuous on D, then f is continuous on \(\Omega _D\).

  • \((\mathrm {ii})\) Let \(k\in \mathbb {N}\setminus \{0\}\). If F is of class \(\mathscr {C}^{4k+3}\) on D, then f is of class \(\mathscr {C}^{k}\) on \(\Omega _D\).

  • \((\mathrm {iii})\) Let \(k\in \{\infty ,\omega \}\). If F is of class \(\mathscr {C}^k\) on D, then f is of class \(\mathscr {C}^k\) on \(\Omega _D\).

Multiplicative structure on slice functions

Every product on \({\mathbb {H}}\otimes \mathbb {R}^4\) induces a product on stem functions, and hence a structure of real algebra on the set of slice functions. We consider the product on \({\mathbb {H}}\otimes \mathbb {R}^4\) constructed as follows. First, we equip \(\mathbb {R}^4\) with the unique (commutative and associative) multiplicative structure which makes the real linear isomorphism \(\mathbb {R}^4\rightarrow \mathbb {C}\otimes \mathbb {C}\), sending \(e_\emptyset \) to the unity \(1=1\otimes 1\), \(e_1\) to \(i\otimes 1\), \(e_2\) to \(1\otimes i\) and \(e_{12}\) to \(i\otimes i\), a real algebra isomorphism. In other words, \(e_\emptyset =1\) is the unity of \(\mathbb {R}^4\), \(e_1^2=e_2^2=-1\) and \(e_1e_2=e_2e_1=e_{12}\). Then, we extend this product to \({\mathbb {H}}\otimes \mathbb {R}^4\) by setting \((a\otimes v)\cdot (b\otimes w)=(ab)\otimes (vw)\) for all \(a,b\in {\mathbb {H}}\) and \(v,w\in \mathbb {R}^4\). In this way, we can identify the real algebra \({\mathbb {H}}\otimes \mathbb {R}^4\) with \({\mathbb {H}}\otimes (\mathbb {C}\otimes \mathbb {C})\).

Definition 1.5

Let \(f,g:\Omega _D \rightarrow {\mathbb {H}}\) be slice functions with \(f=\mathcal {I}(F)\) and \(g=\mathcal {I}(G)\). We define the (tensor) slice product \(f\cdot g:\Omega _D\rightarrow {\mathbb {H}}\) of f and g by \(f\cdot g:=\mathcal {I}(FG)\), where FG is the pointwise product defined by \((FG)(z)=F(z)G(z)\) in \({\mathbb {H}}\otimes (\mathbb {C}\otimes \mathbb {C})\) for all \(z\in D\). \(\square \)

For example, the slice product of the coordinate functions \(x_1:{\mathbb {H}}^2\rightarrow {\mathbb {H}}\) and \(x_2:{\mathbb {H}}^2\rightarrow {\mathbb {H}}\), with \(x_h=\alpha _h+J_h\beta _h=\mathcal {I}(\alpha _h+e_h\beta _h)\) for \(h=1,2\), is the slice function \(x_1\cdot x_2:{\mathbb {H}}^2\rightarrow {\mathbb {H}}\) given by

$$\begin{aligned} x_1\cdot x_2&=\mathcal {I}((\alpha _1+e_1\beta _1)(\alpha _2+e_2\beta _2))=\mathcal {I}(\alpha _1\alpha _2+e_1\alpha _2\beta _1+e_2\alpha _1\beta _2+e_{12}\beta _1\beta _2)\\&=\alpha _1\alpha _2+J_1\alpha _2\beta _1+J_2\alpha _1\beta _2+J_1J_2\beta _1\beta _2. \end{aligned}$$

In this case, the slice product \(x_1\cdot x_2\) coincides with the pointwise product \(x_1x_2\). Moreover, \(x_1\cdot x_2=x_2\cdot x_1\). In general, if \(a,b\in {\mathbb {H}}\), the slice product of \(x_1a=\mathcal {I}(\alpha _1a+e_1(\beta _1a))\) and \(x_2b=\mathcal {I}(\alpha _2b+e_2(\beta _2b))\) is the slice function \((x_1a)\cdot (x_2b)=x_1x_2ab\), while \((x_2b)\cdot (x_1a)=x_1x_2ba\). Note that the pointwise product \(x_2x_1\) is not even a slice function, see Remark 2.14.

The isomorphism from the real algebra of stem functions \(F:D\rightarrow {\mathbb {H}}\otimes (\mathbb {C}\otimes \mathbb {C})\) with the pointwise product to the real algebra of slice functions \(f:\Omega _D\rightarrow {\mathbb {H}}\) with the slice product can be expressed by the commutativity of the following diagrams for all \(J_1,J_2\in {\mathbb {S}}_{\mathbb {H}}\):

figure a

Here \(\phi _{J_1}\times \phi _{J_2}:D\rightarrow \Omega _D\) denotes the product map \((\phi _{J_1}\times \phi _{J_2})(z_1,z_2):=(\phi _{J_1}(z_1),\phi _{J_2}(z_2))\), where \(\phi _J\) is the map defined in (1), and \(\Phi _{J_1,J_2}:{\mathbb {H}}\otimes (\mathbb {C}\otimes \mathbb {C})\rightarrow {\mathbb {H}}\) is the real linear map defined by \(\Phi _{J_1,J_2}(a\otimes (z_1\otimes z_2)):=\phi _{J_1}(z_1)\phi _{J_2}(z_2)a\) for all \(a\in {\mathbb {H}}\) and \(z_1,z_2\in \mathbb {C}\).

Slice regular functions

Let \(\mathcal {J}_1\) and \(\mathcal {J}_2\) be the commuting complex structures on \(\mathbb {R}^4\simeq \mathbb {C}\otimes \mathbb {C}\) induced, respectively, by the standard structures of the two copies of \(\mathbb {C}\). Explicitly, \(\mathcal {J}_h(e_h)=-1\) for \(h=1,2\) and \(\mathcal {J}_1(e_2)=\mathcal {J}_2(e_1)=e_{12}\). We extend these structures to \({\mathbb {H}}\otimes \mathbb {R}^4\) by setting \(\mathcal {J}_h(a\otimes v)=a\otimes \mathcal {J}_h(v)\) for all \(a\in {\mathbb {H}}\) and \(v\in \mathbb {R}^4\).

Throughout the remaining part of this section, we assume that D is open in \(\mathbb {C}^2\).

Definition 1.6

Let \(F:D \rightarrow {\mathbb {H}}\otimes \mathbb {R}^4\) be a stem function of class \(\mathscr {C}^1\). For each \(h=1,2\), we denote \(\partial _h\) and \({{\overline{\partial }}}_h\) the Cauchy-Riemann operators w.r.t. the complex structures i on D and \(\mathcal {J}_h\) on \({\mathbb {H}}\otimes \mathbb {R}^{4}\), i.e.

$$\begin{aligned} \partial _hF=\frac{1}{2}\left( \dfrac{\partial F}{\partial \alpha _h}-\mathcal {J}_h\left( \dfrac{\partial F}{\partial \beta _h}\right) \right) \,\,\text { and }\,\, \quad {{\overline{\partial }}}_hF=\frac{1}{2}\left( \dfrac{\partial F}{\partial \alpha _h}+\mathcal {J}_h\left( \dfrac{\partial F}{\partial \beta _h}\right) \right) , \end{aligned}$$

where \(\alpha _h+i \beta _h:D\rightarrow \mathbb {C}\) is the \(h^{\mathrm {th}}\)-coordinate function of D. Let \(f=\mathcal {I}(F):\Omega _D\rightarrow {\mathbb {H}}\) and let \(h=1,2\). We define the slice partial derivatives of f as the following slice functions on \(\Omega _D\):

$$\begin{aligned} \dfrac{\partial f}{\partial x_h}:=\mathcal {I}(\partial _hF) \,\,\text { and }\,\, \dfrac{\partial f}{\partial x_h^c}:=\mathcal {I}({{\overline{\partial }}}_hF).\, \end{aligned}$$

\(\square \)

We denote \(\mathcal {S}^1(\Omega _D,{\mathbb {H}})\) the set of all slice functions induced by stem functions of class \(\mathscr {C}^1\).

Proposition 1.7

(Leibniz’s rule) For each slice functions \(f,g\in \mathcal {S}^1(\Omega _D,{\mathbb {H}})\) and \(h=1,2\), it holds:

$$\begin{aligned} \frac{\partial }{\partial x_h}(f\cdot g)&=\frac{\partial f}{\partial x_h}\cdot g+f\cdot \frac{\partial g}{\partial x_h},\\ \frac{\partial }{\partial x_h^c}(f\cdot g)&=\frac{\partial f}{\partial x_h^c}\cdot g+f\cdot \frac{\partial g}{\partial x_h^c}. \end{aligned}$$

Definition 1.8

Let \(F:D\rightarrow {\mathbb {H}}\otimes \mathbb {R}^4\) be a stem function of class \(\mathscr {C}^1\) and let \(f=\mathcal {I}(F):\Omega _D \rightarrow {\mathbb {H}}\) be the induced slice function. F is called holomorphic if \({{\overline{\partial }}}_1F={{\overline{\partial }}}_2F=0\) on D. If F is holomorphic, then we say that \(f=\mathcal {I}(F)\) is a slice regular function. \(\square \)

Thanks to Proposition 1.4, every slice regular function is real analytic on \(\Omega _D\).

Example 1.9

All polynomial functions \(f:{\mathbb {H}}^2\rightarrow {\mathbb {H}}\) of the form \(f(x)=\sum _{(\ell _1,\ell _2)\in L}x_1^{\ell _1} x_2^{\ell _2} a_{\ell _1,\ell _2}\), for some finite subset L of \(\mathbb {N}^2\) and \(a_{\ell _1,\ell _2}\in {\mathbb {H}}\), are slice regular. More generally, the sum of a convergent power series \(\sum _{(\ell _1,\ell _2)\in \mathbb {N}^2}x_1^{\ell _1} x_2^{\ell _2} a_{\ell _1,\ell _2}\) is slice regular on a product of two open balls of \({\mathbb {H}}\) centered at the origin. \(\square \)

For each \(J\in {\mathbb {S}}_{\mathbb {H}}\) and for each slice function \(f:\Omega _D\rightarrow {\mathbb {H}}\), we define \(\Omega _D(J):=\Omega _D\cap (\mathbb {C}_J\times \mathbb {C}_J)\) and we denote \(f_J:\Omega _D(J)\rightarrow {\mathbb {H}}\) the restriction of f to \(\Omega _D(J)\).

Proposition 1.10

Let \(f\in \mathcal {S}^1(\Omega _D,{\mathbb {H}})\). The following assertions are equivalent:

  • \((\mathrm {i})\) f is slice regular.

  • \((\mathrm {ii})\) \(\displaystyle \frac{\partial f}{\partial x_h^c}=0\) on \(\Omega _D\) for \(h=1,2\).

  • \((\mathrm {iii})\) There exists \(J \in {\mathbb {S}}_{\mathbb {H}}\) such that \(f_J:\Omega _D(J)\rightarrow {\mathbb {H}}\) is holomorphic w.r.t. the complex structures on \(\Omega _D(J)\) and on \({\mathbb {H}}\) defined by the left multiplication by J; that is,

    $$\begin{aligned} \dfrac{\partial f_J}{\partial \alpha _h}(z)+J\dfrac{\partial f_J}{\partial \beta _h}(z)=0\,\text { for all }z\in \Omega _D(J)\text { and for all }h=1,2, \end{aligned}$$
    (2)

    where \(z=(\alpha _1+J\beta _1,\alpha _2+J\beta _2)\in \mathbb {C}_J\times \mathbb {C}_J\).

  • \((\mathrm {iv})\) For each \(J \in {\mathbb {S}}_{\mathbb {H}}\), \(f_J\) is holomorphic in the sense of (2).

  • \((\mathrm {v})\) f is slice regular w.r.t. \(x_1\), and the spherical value and spherical derivative of f w.r.t. \(x_1\) are slice regular w.r.t. \(x_2\).

As a simple illustration of condition \((\mathrm {v})\), consider the polynomial \(f(x_1,x_2)=x_1x_2\). For every fixed \(x_2\in {\mathbb {H}}\), \(g(x_1):=f(x_1,x_2)=x_1x_2\) is slice regular w.r.t. the variable \(x_1\). Moreover, the spherical value \({g}_s^\circ (x_1)=x_2{\text {Re}}(x_1)\) and the spherical derivative \(g'_s(x_1)=x_2\) of g w.r.t. \(x_1\) are slice regular w.r.t. the variable \(x_2\) for every fixed \(x_1\in {\mathbb {H}}\).

Proposition 1.11

The zero set of any nonconstant polynomial function \(f:{\mathbb {H}}^2\rightarrow {\mathbb {H}}\) is a nonempty real algebraic subset of \(\mathbb {R}^8={\mathbb {H}}^2\), whose real dimension can assume precisely the three values 4, 5 and 6.

Leibniz’s rule (Proposition 1.7) and Proposition 1.10 imply that the slice product preserves slice regularity. In particular, every slice product \(f(x)=f_1(x_1)\cdot f_2(x_2)\), with \(f_h(x_h)\) slice regular w.r.t. the variable \(x_h\) for \(h=1,2\), is slice regular.

Since the complex structures \(\mathcal {J}_1\) and \(\mathcal {J}_2\) commute, every pair of Cauchy-Riemann operators commute. In particular, for every slice regular function f, also the slice partial derivatives \(\frac{\partial f}{\partial x_1}\) and \(\frac{\partial f}{\partial x_2}\) are slice regular.

Cauchy integral formula for slice regular functions

Let \(J\in {\mathbb {S}}_{\mathbb {H}}\) be fixed. Recall that \(\phi _J:\mathbb {C}\rightarrow \mathbb {C}_J\) denotes the real \(^*\)-algebra isomorphism \(\phi _J(\alpha +i\beta ):=\alpha +J\beta \). Let \(E'_1\) and \(E'_2\) be bounded non-empty open subsets of \(\mathbb {C}\) invariant under complex conjugation and with boundaries of class \(\mathscr {C}^1\). Define \(E_h:=\phi _J(E_h')\) for \(h=1,2\), and \(E:=E_1\times E_2\subset \mathbb {C}_J\times \mathbb {C}_J\) with distinguished boundary \(\partial ^*E:=(\partial E_1)\times (\partial E_2)\). Let \(\Omega (E)=\Omega _{E'_1\times E'_2}\) be the circular open subset of \({\mathbb {H}}^2\) such that \(\Omega (E)\cap (\mathbb {C}_J\times \mathbb {C}_J)=E\).

Definition 1.12

We define the slice Cauchy kernel for E as the function \(C:\Omega (E)\times \partial ^*E\rightarrow {\mathbb {H}}\) given by

$$\begin{aligned} C(x,y):=C_{\mathbb {H}}(x_1,y_1)\cdot _x C_{\mathbb {H}}(x_2,y_2), \end{aligned}$$

where each \(C_{\mathbb {H}}(x_h,y_h)=(y_h-x_h)^{-\bullet }=\Delta _{y_h}(x_h)^{-1}(y_h^c-x_h)=(x_h^2-x_ht(y_h)+n(y_h))^{-1}(y_h^c-x_h)\) is the usual slice Cauchy kernel in one quaternionic variable, and the slice product \(\cdot _x \) is performed w.r.t. \(x=(x_1,x_2)\in \Omega (E)\) for each \(y=(y_1,y_2)\in \partial ^*E\). \(\square \)

It is worth noting that \(C(\cdot ,y)\) is slice regular on \(\Omega (E)\) for each fixed \(y\in \partial ^*E\).

For \(h=1,2\), let \(\xi _h:T_h\rightarrow \partial E_h\) be piecewise \(\mathscr {C}^1\) parametrizations of \(\partial E_h\) and let \(T:=T_1\times T_2\). Given two continuous functions \(p,q:\partial ^*E\rightarrow {\mathbb {H}}\), we define

$$\begin{aligned} \int _{\partial ^*E}p(y)\,dy\,q(y):=\int _Tp(\xi _1(t_1),\xi _2(t_2))\,{\dot{\xi }}_1(t_1)\,{\dot{\xi }}_2(t_2)\,q(\xi _1(t_1),\xi _2(t_2))\,dt_1dt_2, \end{aligned}$$

where \({\dot{\xi }}_h:T_h\rightarrow {\mathbb {H}}\) denotes the a.e. defined derivatives of \(\xi _h\).

Theorem 1.13

Let \(f:\Omega _D\rightarrow {\mathbb {H}}\) be a slice regular function. Suppose that the closure of \(\Omega (E)\) in \({\mathbb {H}}^2\) is contained in \(\Omega _D\). Then

$$\begin{aligned} f(x)=(2\pi )^{-2}\int _{\partial ^*E}C(x,y)J^{-2}\,dy\, f(y)\quad \text { for all }x\in \Omega (E), \end{aligned}$$

and the slice Cauchy kernel C can be expressed in terms of pointwise operations as follows:

$$\begin{aligned} C(x,y)=&\,\Delta _{y_1}(x_1)^{-1}x_1\Delta _{y_2}(x_2)^{-1}x_2-\Delta _{y_1}(x_1)^{-1}\Delta _{y_2}(x_2)^{-1}x_2y_1^c+\\&\quad -\Delta _{y_1}(x_1)^{-1}x_1\Delta _{y_2}(x_2)^{-1}y_2^c+\Delta _{y_1}(x_1)^{-1}\Delta _{y_2}(x_2)^{-1}y_1^cy_2^c\\ =&\,\Delta _{y_1}(x_1)^{-1}\big (x_1\Delta _{y_2}(x_2)^{-1}x_2-\Delta _{y_2}(x_2)^{-1}x_2y_1^c-x_1\Delta _{y_2}(x_2)^{-1}y_2^c\\&\quad +\Delta _{y_2}(x_2)^{-1}y_1^cy_2^c\big ) \end{aligned}$$

for all \((x,y)\in \Omega (E)\times \partial ^*E\).

Slice functions

Basic definitions

If S is a set, then we denote |S| the cardinality of S and \(\mathcal {P}(S)\) the set of all subsets of S. Let \(\mathbb {N}\) be the set of all non-negative integers and let \(\mathbb {N}^*:=\mathbb {N}\setminus \{0\}\). For simplicity, given any \(n\in \mathbb {N}^*\), we use the symbol \(\mathcal {P}(n)\) instead of \(\mathcal {P}(\{1,\ldots ,n\})\). Given \(z=(z_1,\ldots ,z_n) \in \mathbb {C}^n\) and \(h \in \{1,\ldots ,n\}\), we define

$$\begin{aligned} {\overline{z}}^h:=(z_1,\ldots ,z_{h-1},\overline{z_h},z_{h+1},\ldots ,z_n), \end{aligned}$$

where \(\overline{z_h}\) denotes the usual conjugation of the complex number \(z_h\). Given a subset D of \(\mathbb {C}^n\), we say that D is invariant under complex conjugations if \({\overline{z}}^h \in D\) for all \(z \in D\) and for all \(h \in \{1,\ldots ,n\}\).

Assumption 2.1

Throughout the paper, A is a real alternative algebra of finite dimension with unity \(1\ne 0\), which we equip with its natural Euclidean topology and structure of real analytic manifold, as a finite dimensional real vector space.

We denote n a positive integer and \(\{e_K\}_{K \in \mathcal {P}(n)}\) a fixed basis of the real vector space \(\mathbb {R}^{2^n}\). We identify \(\mathbb {R}\) with a real vector subspace of \(\mathbb {R}^{2^n}\) via the real linear embedding sending \(1\in \mathbb {R}\) into \(e_\emptyset \in \mathbb {R}^{2^n}\), and we write \(e_{\emptyset }=1\). For simplicity, we set \(e_k:=e_{\{k\}}\) for all \(k\in \{1,\ldots ,n\}\).

We assume that \({\mathbb {S}}_A\ne \emptyset \).

We also assume that D is a non-empty subset of \(\mathbb {C}^n\) invariant under complex conjugations.

Consider the (real) tensor product \(A \otimes \mathbb {R}^{2^n}\). Each element x of \(A \otimes \mathbb {R}^{2^n}\) can be uniquely written as \(x=\sum _{K\in \mathcal {P}(n)}e_Ka_K\) with \(a_K\in A\). In particular, we can identify each element a of A with the element \(e_\emptyset a=1a\) of \(A\otimes \mathbb {R}^{2^n}\), and we write \(a=1a\). As a consequence, A turns out to be a real vector subspace of \(A\otimes \mathbb {R}^{2^n}\). Note that, given any function \(F:D \rightarrow A \otimes \mathbb {R}^{2^n}\), there exist, and are unique, functions \(F_K:D\rightarrow A\) such that \(F=\sum _{K \in \mathcal {P}(n)}e_KF_K\). We shall say that \(F_K\) is the K-component of F.

Definition 2.2

We say that a function \(F:D \rightarrow A \otimes \mathbb {R}^{2^n}\) with \(F=\sum _{K \in \mathcal {P}(n)}e_KF_K\) is a stem function if \(F_K({\overline{z}}^h)=(-1)^{|K \cap \{h\}|}F_K(z)\) or, equivalently,

$$\begin{aligned} F_K({\overline{z}}^h)= \left\{ \begin{array}{ll} F_K(z) &{} \text { if }\,h \not \in K \\ -F_K(z) &{} \text { if }\,h \in K \end{array} \right. \end{aligned}$$
(3)

for all \(z \in D\), \(K \in \mathcal {P}(n)\) and \(h \in \{1,\ldots ,n\}\). We denote \(\mathrm {Stem}(D,A\otimes \mathbb {R}^{2^n})\) the set of all stem functions from D to \(A\otimes \mathbb {R}^{2^n}\). \(\square \)

Let \(z=(z_1,\ldots ,z_n) \in \mathbb {C}^n\) and let \(H=\{h_1,\ldots ,h_p\} \in \mathcal {P}(n)\setminus \{\emptyset \}\) with \(h_1<\ldots <h_p\). Define \({\overline{z}}^{H} \in \mathbb {C}^n\) by setting

$$\begin{aligned} {\overline{z}}^{H}:=(z_1,\ldots ,z_{h_1-1},\overline{z_{h_1}},z_{h_1+1}, \ldots ,z_{h_p-1},\overline{z_{h_p}},z_{h_p+1},\ldots ,z_n). \end{aligned}$$

If \(H=\emptyset \), then we set \({\overline{z}}^H:=z\). Note that \({\overline{z}}^{\{h\}}={\overline{z}}^h\) for all \(h \in \{1,\ldots ,n\}\) and \({\overline{z}}^H \in D\) for all \(z \in D\) and \(H \in \mathcal {P}(n)\). Moreover, if \(F:D \rightarrow A \otimes \mathbb {R}^{2^n}\) is a function with \(F=\sum _{K \in \mathcal {P}(n)}e_KF_K\), then F is a stem function if and only if

$$\begin{aligned} F_K({{\bar{z}}}^{H})=(-1)^{|K\cap H|}F_K(z) \, \text { for all }z\in D\text { and }K,H \in \mathcal {P}(n). \end{aligned}$$
(4)

Definition 2.3

Let W be a subset of \(\mathbb {C}^n\). We call circularization of W (in \(A^n\)) the set \(\Omega _W\) of all points \((\alpha _1+J_1\beta _1,\ldots ,\alpha _n+J_n\beta _n)\in (Q_A)^n\) with \(\alpha _1,\beta _1,\ldots ,\alpha _n,\beta _n \in \mathbb {R}\) and \(J_1,\ldots ,J_n \in {\mathbb {S}}_A\) such that \((\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n) \in W\). A subset \(\Theta \) of \((Q_A)^n\) is said to be circular (in \(A^n\)) if \(\Theta =\Omega _W\) for some subset W of \(\mathbb {C}^n\).

Given any \(x\in (Q_A)^n\), we denote \({\mathbb {S}}_x\) the smallest circular subset of \((Q_A)^n\) containing x.

\(\square \)

Note that, if \(x=(x_1,\ldots ,x_n)\in (Q_A)^n\), then \({\mathbb {S}}_x={\mathbb {S}}_{x_1}\times \cdots \times {\mathbb {S}}_{x_n}\), where \({\mathbb {S}}_{x_h}=\alpha _h+{\mathbb {S}}_A\beta _h\) if \(x_h=\alpha _h+J_h\beta _h\) for some \(\alpha _h,\beta _h\in \mathbb {R}\) and \(J_h\in {\mathbb {S}}_A\). Given another point \(y=(y_1,\ldots ,y_n)\) in \((Q_A)^n\), we have that \({\mathbb {S}}_x={\mathbb {S}}_y\) if and only if \(t(x_h)=t(y_h)\) and \(n(x_h)=n(y_h)\) for all \(h\in \{1,\ldots ,n\}\).

Let us introduce a notation, which is very useful especially in the non-associative case.

Definition 2.4

Given any \(m\in \mathbb {N}^*\) and any sequence \(u=(u_1,\ldots ,u_m)\) of elements of A, we define the ordered product \([u]=[u_1,\ldots ,u_m]\) of u by setting \([u]=[u_1,\ldots ,u_m]:=u_1\) if \(m=1\) and

$$\begin{aligned} {[}u]=[u_1,\ldots ,u_m]:=u_1(u_2(u_3\cdots (u_{m-1}u_m)\ldots )) \end{aligned}$$

if \(m\ge 2\). Given any \(v\in A\), we write [uv] to indicate \([u_1,\ldots ,u_m,v]\).

Moreover, we set \([\emptyset ]:=1\) and \([\emptyset ,v]:=v\).

Let \(H\in \mathcal {P}(m)\). If \(H=\emptyset \), then we define \(u_H:=\emptyset \); hence \([u_H]=1\) and \([u_H,v]=v\). If \(H\ne \emptyset \), then we write \(H=\{h_1,\ldots ,h_p\}\) with \(h_1<\ldots <h_p\), and we define \(u_H:=(u_{h_1},\ldots ,u_{h_p})\); as a consequence, we have:

$$\begin{aligned} {[}u_H]=u_{h_1}(u_{h_2}(u_{h_3}\cdots (u_{h_{p-1}}u_{h_p})\ldots )) \end{aligned}$$

and

$$\begin{aligned} {[}u_H,v]=u_{h_1}(u_{h_2}(u_{h_3}\cdots (u_{h_{p-1}}(u_{h_p}v))\ldots )). \end{aligned}$$

We use also the symbols \((u_h)_{h=1}^m\) to denote u, and \((u_h)_{h\in H}\) to denote \(u_H\).

Suppose now that, for each \(h\in \{1,\ldots ,m\}\), \(u_h\) is invertible in A, and denote \(u_h^{-1}\) its inverse \((u_h)^{-1}\) in A. In this case, if \(H=\emptyset \), then we define \(u_H^{-1}:=\emptyset \); hence \([u_H^{-1}]=1\) and \([u_H^{-1},v]=v\). If \(H\ne \emptyset \) and \(H=\{h_1,\ldots ,h_p\}\) with \(h_1<\ldots <h_p\), then we define \(u_H^{-1}:=(u_{h_p}^{-1},u_{h_{p-1}}^{-1},\ldots ,u_{h_1}^{-1})\); as a consequence, we have:

$$\begin{aligned} {[}u_H^{-1}]=u_{h_p}^{-1}(u_{h_{p-1}}^{-1}(u_{h_{p-2}}^{-1}\cdots (u_{h_2}^{-1}u_{h_1}^{-1})\ldots )) \end{aligned}$$

and

$$\begin{aligned} {[}u_H^{-1},v]=u_{h_p}^{-1}(u_{h_{p-1}}^{-1}(u_{h_{p-2}}^{-1}\cdots (u_{h_2}^{-1}(u_{h_1}^{-1}v))\ldots )). \end{aligned}$$

We use also the symbols \(((u_h)_{h=1}^m)^{-1}\) to denote \((u_{m+1-h}^{-1})_{h=1}^m\), and \(((u_h)_{h\in H})^{-1}\) to denote \(u_H^{-1}\). \(\square \)

Given any \(v,w\in A\), any \(H\in \mathcal {P}(m)\), and any \(u=(u_1,\ldots ,u_m)\in A^m\) with \(m\ge 1\) and \(u_1,\ldots ,u_m\) invertible in A, it is immediate to verify that

$$\begin{aligned}{}[u_H,v]=w\, \text { if and only if }\,v=[u_H^{-1},w]. \end{aligned}$$
(5)

The elements \([u_H]\) and \([u_H^{-1}]\) are invertible in A, see for instance Lemma 1.5(2) of [13]. However, in general, if A is not associative, then \([u_H^{-1}]\ne [u_H]^{-1}\). On the contrary, if A is associative, then \([u_H^{-1}]=[u_H]^{-1}\) and \([u_H]=u_{h_1}u_{h_2}\cdots u_{h_p}\).

We are in position to introduce the notion of slice function in several variables.

Definition 2.5

Given a function \(f:\Omega _D\rightarrow A\), we say that f is a (left) slice function if there exists a stem function \(F:D \rightarrow A \otimes \mathbb {R}^{2^n}\) with \(F=\sum _{K \in \mathcal {P}(n)}e_KF_K\) such that

$$\begin{aligned} \textstyle f(x):=\sum _{K \in \mathcal {P}(n)}[J_K,F_K(z)] \end{aligned}$$

for all \(x=(\alpha _1+J_1\beta _1,\ldots ,\alpha _n+J_n\beta _n) \in \Omega _D\), where \(\alpha _1,\beta _1,\ldots ,\alpha _n,\beta _n \in \mathbb {R}\), \(z=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\in D\) and \(J=(J_1,\ldots ,J_n) \in ({\mathbb {S}}_A)^n\); hence \(J_K=(J_{k_1},\ldots ,J_{k_p})\) if \(K=\{k_1,\ldots ,k_p\}\in \mathcal {P}(n)\setminus \{\emptyset \}\) with \(k_1<\cdots <k_p\), and \(J_\emptyset =\emptyset \).

If this is the case, we say that f is induced by F, and we write \(f=\mathcal {I}(F)\). We denote \(\mathcal {S}(\Omega _D,A)\) the set of all slice functions from \(\Omega _D\) to A, and \(\mathcal {I}:\mathrm {Stem}(D,A\otimes \mathbb {R}^{2^n})\rightarrow \mathcal {S}(\Omega _D,A)\) the map sending each stem function F into the corresponding slice function \(\mathcal {I}(F)\). \(\square \)

The preceding definition is well-posed. Let \(x=(x_1,\ldots ,x_n)\) be a point of \(\Omega _D\). For each \(h\in \{1,\ldots ,n\}\), there exist \(\alpha _h,\beta _h\in \mathbb {R}\) with \(\beta _h\ge 0\) and \(J_h\in {\mathbb {S}}_A\) such that \(x_h=\alpha _h+J_h\beta _h\). If \(x_h\in \mathbb {R}\), i.e., \(\beta _h=0\), then \(\alpha _h\) is uniquely determined by x; on the contrary, \(J_h\) can be chosen arbitrarily in \({\mathbb {S}}_A\). If \(x_h\not \in \mathbb {R}\), i.e. \(\beta _h>0\), then \(\alpha _h\), \(\beta _h\) and \(J_h\) are uniquely determined by \(x_h\), and \(x_h\) has the following two representations:

$$\begin{aligned} \alpha _h+J_h\beta _h=x_h=\alpha _h+(-J_h)(-\beta _h). \end{aligned}$$

Set \(z:=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\in D\). Let L be the set of all \(h\in \{1,\ldots ,n\}\) such that \(x_h\not \in \mathbb {R}\). Thanks to (3), we know that \(F_K(z)=0\) if \(K\not \subset L\). For each \(H\in \mathcal {P}(n)\) with \(\emptyset \ne H\subset L\), it is possible to write x as follows:

$$\begin{aligned} x=(\alpha _1+J_1\beta _1,\ldots ,\alpha _n+J_n\beta _n) \end{aligned}$$

and \(x=x'_H\), where

$$\begin{aligned} x'_H=(\alpha _1+(\epsilon _1 J_1)(\epsilon _1\beta _1),\ldots ,\alpha _n+(\epsilon _n J_n)(\epsilon _n\beta _n)) \end{aligned}$$

with \(\epsilon _h=-1\) if \(h\in H\), \(\epsilon _h=1\) if \(h\in L\setminus H\), and \((\epsilon _h,J_h)\) can be chosen arbitrarily in \(\{-1,1\}\times {\mathbb {S}}_A\) if \(h\not \in L\). By (4), we have

$$\begin{aligned} f(x)&\textstyle =\sum _{K \in \mathcal {P}(n)}[J_K,F_K(z)]=\sum _{K \in \mathcal {P}(n),K\subset L}[J_K,F_K(z)]\\&\textstyle =\sum _{K \in \mathcal {P}(n),K\subset L}[J_K,(-1)^{|K\cap H|}F_K({\overline{z}}^H)]\\&\textstyle =\sum _{K \in \mathcal {P}(n)}[((-1)^{|H\cap \{k\}|}J_k)_{k\in K},F_K({\overline{z}}^H)]=f(x'_H). \end{aligned}$$

It follows that Definition 2.5 is well-posed, as claimed. In Proposition 2.12 below, we will show that a slice function is induced by a unique stem function.

The real algebra A we are working with is assumed to be alternative. In particular, it is power-associative. Consequently, if \(a\in A\) and \(m\in \mathbb {N}^*\) then the power \(a^m\) is a well-defined element of A, independently from the system of parentheses we use to compute it. For convention, we set \(a^0:=1\) for all \(a\in A\).

Remark 2.6

The usual pointwise defined operations of addition and multiplication by real scalars define structures of real vector spaces on the sets \(\mathrm {Stem}(D,A\otimes \mathbb {R}^{2^n})\) and \(\mathcal {S}(\Omega _D,A)\), which make the map \(\mathcal {I}:\mathrm {Stem}(D,A\otimes \mathbb {R}^{2^n})\rightarrow \mathcal {S}(\Omega _D,A)\) a real linear map. Given any \(F=\sum _{K\in \mathcal {P}(n)}e_KF_K,G=\sum _{K\in \mathcal {P}(n)}e_KG_K\in \mathrm {Stem}(D,A\otimes \mathbb {R}^{2^n})\), \(f,g\in \mathcal {S}(\Omega _D,A)\) and \(r\in \mathbb {R}\), we set \((F+G)(z):=\sum _{K\in \mathcal {P}(n)}e_K(F_K(z)+G_K(z))\), \((Fr)(z):=\sum _{K\in \mathcal {P}(n)}e_K(F_K(z)r)\), \((f+g)(x):=f(x)+g(x)\) and \((fr)(x):=f(x)r\), where \(F_K(z)+G_K(z)\), \(F_K(z)r\), \(f(x)+g(x)\) and f(x)r are additions and scalar multiplications in A. Evidently, \(\mathcal {I}(F+G)=\mathcal {I}(F)+\mathcal {I}(G)\) and \(\mathcal {I}(Fr)=\mathcal {I}(F)r\). Actually, the map \(\mathcal {I}\) is an isomorphism of real vector spaces, see Corollary 2.15 below. \(\square \)

Definition 2.7

Given \(x=(x_1,\ldots ,x_n)\in A^n\), \(\ell =(\ell _1,\ldots ,\ell _n)\in \mathbb {N}^n\) and \(a\in A\), we denote \(x^\ell a\) the element \([(x_h^{\ell _h})_{h=1}^n,a]\) of A. A function \(P:(Q_A)^n\rightarrow A\) is monomial if there exist \(\ell \in \mathbb {N}^n\) and \(a\in A\) such that \(P(x)=x^\ell a\) for all \(x\in (Q_A)^n\). The function \(P:(Q_A)^n\rightarrow A\) is polynomial if it is a finite sum of monomial functions.

The restriction of a monomial (respectively polynomial) function to \(\Omega _D\) is said to be monomial (respectively polynomial) on \(\Omega _D\). \(\square \)

We conclude the present section with an important result, which asserts that the class of slice functions includes the one of polynomial functions. First, we need a definition.

Definition 2.8

For each \(k\in \mathbb {N}\), we denote \(p_k\) and \(q_k\) the real polynomials in \(\mathbb {R}[X,Y]\) such that \((\alpha +i\beta )^k=p_k(\alpha ,\beta )+iq_k(\alpha ,\beta )\) for all \(\alpha ,\beta \in \mathbb {R}\). \(\square \)

Proposition 2.9

Each polynomial function on \(\Omega _D\) is a slice function. More precisely, given any \(\ell =(\ell _1,\ldots ,\ell _n)\in \mathbb {N}^n\) and \(a\in A\), the function \(F^{(\ell )}:D\rightarrow A\otimes \mathbb {R}^{2^n}\), defined by

$$\begin{aligned} \textstyle F^{(\ell )}(z)=\sum _{K\in \mathcal {P}(n)}e_K\big (\big (\prod _{h\in \{1,\ldots ,n\}\setminus K}p_{\ell _h}(\alpha _h,\beta _h)\big )\big (\prod _{h\in K}q_{\ell _h}(\alpha _h,\beta _h)\big )a\big ), \end{aligned}$$
(6)

for all \(z=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\in D\), is a stem function inducing the monomial function \(x^\ell a\) on \(\Omega _D\).

Proof

Let \(\ell =(\ell _1,\ldots ,\ell _n)\in \mathbb {N}^n\). It suffices to show that the function \(F^{(\ell )}:D\rightarrow A\), defined in (6), is a stem function and \(\mathcal {I}(F^{(\ell )})(x)=x^\ell a_\ell \). Let \(z=(z_1,\ldots ,z_n)=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\in D\), let \(h\in \{1,\ldots ,n\}\) and, given any \(K\in \mathcal {P}(n)\), let \(F_K^{(\ell )}:D\rightarrow A\) be the function

$$\begin{aligned} \textstyle F_K^{(\ell )}(z):=\big (\prod _{h\in \{1,\ldots ,n\}\setminus K}p_{\ell _h}(\alpha _h,\beta _h)\big )\big (\prod _{h\in K}q_{\ell _h}(\alpha _h,\beta _h)\big )a. \end{aligned}$$

Note that, for each \(z_h=\alpha _h+i\beta _h\in \mathbb {C}\), it holds:

$$\begin{aligned} p_{\ell _h}(\alpha _h,\beta _h)-iq_{\ell _h}(\alpha _h,\beta _h)=\overline{z_h^{\ell _h}}=\overline{z_h}^{\ell _h}=p_{\ell _h}(\alpha _h,-\beta _h)+iq_{\ell _h}(\alpha _h,-\beta _h) \end{aligned}$$

and hence \(p_{\ell _h}(\alpha _h,-\beta _h)=p_{\ell _h}(\alpha _h,\beta _h)\) and \(q_{\ell _h}(\alpha _h,-\beta _h)=-q_{\ell _h}(\alpha _h,\beta _h)\). Consequently, we have \(F_K^{(\ell )}({\overline{z}}^h)=-F_K^{(\ell )}(z)\) if \(h\in K\) and \(F_K^{(\ell )}({\overline{z}}^h)=F_K^{(\ell )}(z)\) if \(h\in \{1,\ldots ,n\}\setminus K\). This proves that \(F^{(\ell )}=\sum _{K\in \mathcal {P}(n)}e_KF^{(\ell )}_K\) is a stem function. Moreover, if \(x=(\alpha _1+J_1\beta _1,\ldots ,\alpha _n+J_n\beta _n) \in \Omega _D\) for some \(J_1,\ldots ,J_n \in {\mathbb {S}}_A\), then

$$\begin{aligned} \mathcal {I}(F^{(\ell )})(x)&\textstyle =\sum _{K\in \mathcal {P}(n)}[(J_h)_{h\in K},F_K^{(\ell )}(z)]\\&\textstyle =\sum _{K\in \mathcal {P}(n)}\big (\prod _{h\in \{1,\ldots ,n\}\setminus K}p_{\ell _h}(\alpha _h,\beta _h)\big )[(J_hq_{\ell _h}(\alpha _h,\beta _h))_{h\in K},a]\\&\textstyle =[(p_{\ell _h}(\alpha _h,\beta _h)+J_hq_{\ell _h}(\alpha _h,\beta _h))_{h=1}^n,a]=[(x_h^{\ell _h})_{h=1}^n,a]=x^\ell a. \end{aligned}$$

The proof is complete. \(\square \)

Definition 2.10

We say that a stem function \(F:\mathbb {C}^n\rightarrow A\otimes \mathbb {R}^{2^n}\) is monomial, or polynomial, if \(\mathcal {I}(F):(Q_A)^n\rightarrow A\) is. The restriction of a monomial (respectively polynomial) stem function to D is said to be monomial (respectively polynomial) on D. \(\square \)

Thanks to Proposition 2.12 below, a stem function \(F:D\rightarrow A\otimes \mathbb {R}^{2^n}\) is monomial on D if and only if it has form (6).

Representation formulas

We need an elementary, but useful, combinatorial lemma.

Lemma 2.11

For each \(H,L \in \mathcal {P}(n)\), it holds:

$$\begin{aligned} \textstyle \sum _{K \in \mathcal {P}(n)}(-1)^{|H \cap K|+|K \cap L|}=2^n\delta _{H,L}\,, \end{aligned}$$

where \(\delta _{H,L}=1\) if \(H=L\) and \(\delta _{H,L}=0\) otherwise.

Proof

Since \(|H \cap K|+|K \cap L|=|K \cap (H \setminus L)|+|K \cap (L \setminus H)|+2|K \cap H \cap L|\) for all \(K \in \mathcal {P}(n)\), it suffices to prove that the sum \(s(H,L):=\sum _{K \in \mathcal {P}(n)}(-1)^{|K \cap (H \setminus L)|+|K \cap (L \setminus H)|}\) is equal to \(2^n\) if \(H=L\) and is null otherwise. This is evident in the case in which \(L=H\). Let \(L \ne H\). If \(H \subset L\), then we have:

$$\begin{aligned} \textstyle s(H,L)= & {} \textstyle \sum _{K \in \mathcal {P}(n)}(-1)^{|K \cap (L \setminus H)|}=\sum _{S_1 \in \mathcal {P}(L \setminus H), \, S_2 \in \mathcal {P}(\complement (L \setminus H))}(-1)^{|S_1|}\\= & {} \textstyle 2^{n-|L \setminus H|}\sum _{S_1 \in \mathcal {P}(L \setminus H)}(-1)^{|S_1|}=2^{n-|L \setminus H|}\sum _{h=0}^{|L \setminus H|}{|L \setminus H| \atopwithdelims ()h}(-1)^h\\= & {} \textstyle 2^{n-|L \setminus H|}(1+(-1))^{|L \setminus H|}=0, \end{aligned}$$

where \(\complement (L \setminus H)\) is the complement of \(L \setminus H\) in \(\{1,\ldots ,n\}\). Similarly, one proves that \(s(H,L)=0\) if \(L \subset H\). Finally, suppose \(H \not \subset L\) and \(L \not \subset H\). We have:

$$\begin{aligned} \textstyle s(H,L)= & {} \textstyle \sum _{K \in \mathcal {P}(n)}(-1)^{|K \cap (L \setminus H)|+|K \cap (H \setminus L)|}\\= & {} \textstyle \sum _{S_1 \in \mathcal {P}(L \setminus H), \, S_2 \in \mathcal {P}(H \setminus L), \, S_3\in \mathcal {P}(\complement (L \bigtriangleup H))}(-1)^{|S_1|}(-1)^{|S_2|}\\= & {} \textstyle 2^{n-|L \bigtriangleup H|}\big (\sum _{S_1 \in \mathcal {P}(L \setminus H)}(-1)^{|S_1|}\big )\big (\sum _{S_2 \in \mathcal {P}(H \setminus L)}(-1)^{|S_2|}\big )\\= & {} \textstyle 2^{n-|L \bigtriangleup H|}(1+(-1))^{|L \setminus H|}(1+(-1))^{|H \setminus L|}=0, \end{aligned}$$

where \(L \bigtriangleup H\) is the usual symmetric difference between L and H. \(\square \)

Let \(x=(x_1,\ldots ,x_n)\in A^n\) and let \(H \in \mathcal {P}(n)\). If \(H=\emptyset \), then we set \(x^{c,H}:=x\). Suppose that \(H\ne \emptyset \) and write \(H=\{h_1,\ldots ,h_p\}\) with \(h_1<\ldots <h_p\). We denote \(x^{\, c,H}\) the element of \(A^n\) defined as follows:

$$\begin{aligned} x^{\, c,H}:=(x_1,\ldots ,x_{h_1-1},x_{h_1}^{\, c},x_{h_1+1},\ldots ,x_{h_p-1},x_{h_p}^{\, c},x_{h_p+1},\ldots ,x_n). \end{aligned}$$

We have the following representation formulas.

Proposition 2.12

(Representation formula) Let \(f:\Omega _D \rightarrow A\) be a slice function and let \(y \in \Omega _D\). Write y as follows:

$$\begin{aligned} y=(\alpha _1+I_1\beta _1,\ldots ,\alpha _n+I_n\beta _n), \end{aligned}$$

where \(\alpha _h,\beta _h \in \mathbb {R}\) and \(I_h \in \mathbb {S}_A\) for all \(h \in \{1,\ldots ,n\}\). Then it holds:

$$\begin{aligned} \textstyle f(x)=2^{-n}\sum _{K,H \in \mathcal {P}(n)}(-1)^{|K \cap H|}[J_K,[I_K^{-1}, f(y^{\, c,H})]], \end{aligned}$$
(7)

where \(x=(\alpha _1+J_1\beta _1,\ldots ,\alpha _n+J_n\beta _n)\) for some \(J=(J_1,\ldots ,J_n)\in ({\mathbb {S}}_A)^n\), and \(I:=(I_1,\ldots ,I_n)\).

Furthermore, if \(F=\sum _{K \in \mathcal {P}(n)}e_KF_K\) is a stem function inducing f, then we have:

$$\begin{aligned} F_K(z)&\textstyle =2^{-n}[I_K^{-1},\sum _{H \in \mathcal {P}(n)}(-1)^{|K \cap H|}f(y^{\, c,H})]=\nonumber \\&\textstyle =2^{-n}\sum _{H \in \mathcal {P}(n)}(-1)^{|K \cap H|}[I_K^{-1},f(y^{\, c,H})] \end{aligned}$$
(8)

for all \(K \in \mathcal {P}(n)\), where \(z=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n) \in D\). In particular, each slice function f is induced by a unique stem function F.

Proof

Let \(x=(\alpha _1+J_1\beta _1,\ldots ,\alpha _n+J_n\beta _n)\). Define \(z:=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n) \in D\). Let \(K \in \mathcal {P}(n)\). Thanks to (4) and to Lemma 2.11, we obtain:

$$\begin{aligned}&\textstyle \sum _{H \in \mathcal {P}(n)}(-1)^{|K \cap H|}f(y^{\, c,H}) \\&\quad = \textstyle \sum _{H \in \mathcal {P}(n)}(-1)^{|K \cap H|}\big (\sum _{L \in \mathcal {P}(n)}[I_L,F_L({\overline{z}}^H)]\big )\\&\quad = \textstyle \sum _{H \in \mathcal {P}(n)}(-1)^{|K \cap H|}\big (\sum _{L \in \mathcal {P}(n)}(-1)^{|H \cap L|}[I_L,F_L(z)]\big )\\&\quad = \textstyle \sum _{L \in \mathcal {P}(n)}[I_L,F_L(z)]\big (\sum _{H \in \mathcal {P}(n)}(-1)^{|K \cap H|+|H \cap L|}\big )\\&\quad = \textstyle 2^n[I_K,F_K(z)]. \end{aligned}$$

Bearing in mind (5), we deduce (8). Consequently,

$$\begin{aligned} f(x)= & {} \textstyle \sum _{K \in \mathcal {P}(n)}[J_K,2^{-n} \sum _{H \in \mathcal {P}(n)}(-1)^{|K \cap H|}[I_K^{-1},f(y^{\, c,H})]]\\= & {} \textstyle 2^{-n}\sum _{K,H \in \mathcal {P}(n)}(-1)^{|K \cap H|}[J_K,[I_K^{-1},f(y^{\, c,H})]], \end{aligned}$$

as desired. \(\square \)

As an immediate corollary, we obtain:

Corollary 2.13

(Identity principle) Let \(f,g:\Omega _D \rightarrow A\) be slice functions and let \(I_1,\ldots ,I_n \in \mathbb {S}_A\) such that \(f=g\) on \(\Omega _D \cap (\mathbb {C}_{I_1}\times \ldots \times \mathbb {C}_{I_n})\). Then \(f=g\) on the whole \(\Omega _D\).

Remark 2.14

Let \(n\ge 2\) and let \(f:\mathbb {H}^n\rightarrow \mathbb {H}\) be the function \(f(x_1,\ldots ,x_n):=x_2x_1\), i.e. the pointwise product between the coordinate functions \(x_2\) and \(x_1\). The function f is not slice. Otherwise, being \(x_2x_1=x_1x_2\) on \((\mathbb {C}_i)^n\), Proposition 2.9 and Corollary 2.13 would imply that \(x_2x_1=x_1x_2\) on the whole \(\mathbb {H}^n\), i.e. the algebra of quaternions is commutative, which is false. \(\square \)

Bearing in mind Remark 2.6, we have:

Corollary 2.15

The map \(\mathcal {I}:\mathrm {Stem}(D,A\otimes \mathbb {R}^{2^n})\rightarrow \mathcal {S}(\Omega _D,A)\), sending stem functions F into the corresponding slice functions \(\mathcal {I}(F)\), is a bijection, and hence a real linear isomorphism.

Another consequence is the following intrinsic characterization of sliceness.

Corollary 2.16

(Sliceness criterion) Let \(f:\Omega _D \rightarrow A\) be a function. Then f is a slice function if and only if there exist \(I=(I_1,\ldots ,I_n)\in ({\mathbb {S}}_A)^n\) with the following property:

$$\begin{aligned} \textstyle f(x)=2^{-n}\sum _{K,H \in \mathcal {P}(n)}(-1)^{|K \cap H|}[J_K,[I_K^{-1},f(y^{\, c,H})]] \end{aligned}$$
(9)

for all \(y=(\alpha _1+I_1\beta _1,\ldots ,\alpha _n+I_n\beta _n) \in \Omega _D\) and for all \(x=(\alpha _1+J_1\beta _1,\ldots ,\alpha _n+J_n\beta _n) \in \Omega _D\), where \(\alpha _1,\beta _1,\ldots ,\alpha _n,\beta _n \in \mathbb {R}\) and \(J=(J_1,\ldots ,J_n)\in ({\mathbb {S}}_A)^n\).

Proof

If f is slice, then (9) follows from (7).

Suppose that (9) holds for some \(I=(I_1,\ldots ,I_n)\in ({\mathbb {S}}_A)^n\). We will show that \(f=\mathcal {I}(F)\) for some stem function \(F:D \rightarrow A \otimes \mathbb {R}^{2^n}\). For each \(K \in \mathcal {P}(n)\), define the function \(F_K:D \rightarrow A\) by setting

$$\begin{aligned} \textstyle F_K(z):=2^{-n}\sum _{H \in \mathcal {P}(n)}(-1)^{|K \cap H|}[I_K^{-1},f(y^{\, c,H})], \end{aligned}$$

where \(y=(\alpha _1+I_1\beta _1,\ldots ,\alpha _n+I_n\beta _n)\) if \(z=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\in D\).

Fix \(K \in \mathcal {P}(n)\), \(h \in \{1,\ldots ,n\}\) and \(z=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n) \in D\). Set \(y:=(\alpha _1+I_1\beta _1,\ldots ,\alpha _n+I_n\beta _n)\). Note that \(|K \cap H|=|K \cap (H \bigtriangleup \{h\})|-(-1)^{|H \cap \{h\}|}|K \cap \{h\}|\) for all \(H \in \mathcal {P}(n)\). Moreover, the map \(\Psi _h:\mathcal {P}(n) \rightarrow \mathcal {P}(n)\), sending H into \(H \bigtriangleup \{h\}\), is a bijection. Bearing in mind the last two elementary facts and (5), we have that

$$\begin{aligned} \textstyle 2^n[I_K,F_K({\overline{z}}^h)]= & {} \textstyle \sum _{H \in \mathcal {P}(n)}(-1)^{|K \cap H|}f(y^{\, c,H \bigtriangleup \{h\}})\\= & {} \textstyle \sum _{H \in \mathcal {P}(n)}(-1)^{|K \cap (H \bigtriangleup \{h\})|+|K \cap \{h\}|}f(y^{\, c,H \bigtriangleup \{h\}})\\= & {} \textstyle (-1)^{|K \cap \{h\}|}\sum _{H \in \mathcal {P}(n)}(-1)^{|K \cap \Psi _h(H)|}f(y^{\, c,\Psi _h(H)}) \\= & {} \textstyle (-1)^{|K \cap \{h\}|}\sum _{H \in \mathcal {P}(n)}(-1)^{|K \cap H|}f(y^{\, c,H}) \\= & {} \textstyle 2^n(-1)^{|K \cap \{h\}|}[I_K,F_K(z)]; \end{aligned}$$

consequently, \([I_K,F_K({\overline{z}}^h)]=(-1)^{|K \cap \{h\}|}[I_K,F_K(z)]\). Using (5) again, we deduce:

$$\begin{aligned} F_K({\overline{z}}^h)=[I_K^{-1},[I_K,F_K({\overline{z}}^h)]]=(-1)^{|K \cap \{h\}|}[I_K^{-1},[I_K,F_K(z)]]=(-1)^{|K \cap \{h\}|}F_K(z). \end{aligned}$$

In other words, the function \(F:D \rightarrow A \otimes \mathbb {R}^{2^n}\), defined by \(F:=\sum _{K \in \mathcal {P}(n)}e_KF_K\), is a stem function. Formula (9) now ensures that \(f=\mathcal {I}(F)\). \(\square \)

A consequence of the last result is as follows.

Corollary 2.17

Let \(\{f_l:\Omega _D\rightarrow A\}_{l\in \mathbb {N}}\) be a sequence of slice functions, which pointwise converges to a function \(f:\Omega _D\rightarrow A\). Then f is a slice function.

Proof

Let xyIJ be as in the statement of Corollary 2.16. Since each \(f_l\) is a slice function, equation (9) holds for each \(f_l\). Note that \(\{f_l(x)\}_{l\in \mathbb {N}}\) converges to f(x) and \(\{[J_K,[I_K^{-1},f_l(y^{\, c,H})]]\}_{l\in \mathbb {N}}\) converges to \([J_K,[I_K^{-1},f(y^{\, c,H})]]\) for all \(K,H\in \mathcal {P}(n)\). It follows that Eq. (9) holds also for f. Using Corollary 2.16 again, we deduce that f is a slice function. \(\square \)

Spherical value and spherical derivatives

Let \(F=\sum _{K \in \mathcal {P}(n)}e_KF_K\) be a stem function and let \(f:\Omega _D \rightarrow A\) be the slice function \(\mathcal {I}(F)\).

Definition 2.18

We call spherical value of f the slice function \({f}_s^\circ :\Omega _D\rightarrow A\) induced by the A-valued stem function \(F_\emptyset :D\rightarrow A\), that is \({f}_s^\circ :=\mathcal {I}(F_\emptyset )\). \(\square \)

From (8) it follows that

$$\begin{aligned} \textstyle {f}_s^\circ (x)=2^{-n}\sum _{H\in \mathcal {P}(n)}f(x^{c,H}) \end{aligned}$$
(10)

for all \(x\in \Omega _D\). For each \(K\in \mathcal {P}(n)\setminus \{\emptyset \}\), we define:

  • \(D_K:=\bigcap _{k\in K}\{(z_1,\ldots ,z_n)\in D:z_k\not \in \mathbb {R}\}\), assumed to be non-empty.

  • \(F_K^*:D_K\rightarrow A\) by \(F_K^*(z):=\beta _K^{-1}F_K(z)\), where \(z=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\in D_K\) and \(\beta _K:=\prod _{k\in K}\beta _k\).

  • \(\mathbb {R}_K:=\bigcup _{k\in K}\{(x_1,\ldots ,x_n)\in A^n:x_k\in \mathbb {R}\}\).

Note that \(\mathbb {R}_K\) is closed in A, \(D_K\) is invariant under all the complex conjugations of \(\mathbb {C}^n\), and the circularization of \(D_K\) in \(A^n\) is equal to \(\Omega _D\setminus \mathbb {R}_K\), i.e.

$$\begin{aligned} \Omega _{D_K}=\Omega _D\setminus \mathbb {R}_K. \end{aligned}$$

Furthermore, it is immediate to verify that the function \(F_K^*\) is a A-valued stem function on \(D_K\).

Definition 2.19

For each \(K\in \mathcal {P}(n)\setminus \{\emptyset \}\), we call spherical K-derivative of f the slice function \(f'_{s,K}:\Omega _D\setminus \mathbb {R}_K\rightarrow A\) induced by \(F_K^*\), that is \(f'_{s,K}:=\mathcal {I}(F_K^*)\). \(\square \)

Bearing in mind (8) and the equality \({\text {Im}}(x)=\frac{x-x^c}{2}\), given any \(K\in \mathcal {P}(n)\setminus \{\emptyset \}\), we have

$$\begin{aligned} \textstyle f'_{s,K}(x)&\textstyle =2^{-n}[((\beta _kJ_k)_{k\in K})^{-1},\sum _{H\in \mathcal {P}(n)}(-1)^{|K\cap H|}f(x^{c,H})]\\&\textstyle =2^{-n}[(({\text {Im}}(x_k))_{k\in K})^{-1},\sum _{H\in \mathcal {P}(n)}(-1)^{|K\cap H|}f(x^{c,H})] \end{aligned}$$

for all \(x=(x_1,\ldots ,x_n)\in \Omega _D\setminus \mathbb {R}_K\). As a consequence, if for each \(x=(x_1,\ldots ,x_n)\in A^n\) and \(K\in \mathcal {P}(n)\setminus \{\emptyset \}\) we set

$$\begin{aligned} \textstyle {\text {Im}}_K(x):=({\text {Im}}(x_k))_{k\in K}, \end{aligned}$$
(11)

then we have

$$\begin{aligned} \textstyle f'_{s,K}(x)=2^{-n}[({\text {Im}}_K(x))^{-1},\sum _{H\in \mathcal {P}(n)}(-1)^{|K\cap H|}f(x^{c,H})] \end{aligned}$$
(12)

for all \(x\in \Omega _D\setminus \mathbb {R}_K\). Note that the latter equality can be rewritten as follows:

$$\begin{aligned} \textstyle f'_{s,K}(x)=2^{|K|-n}[((x_k-(x_k)^c)_{k\in K})^{-1},\sum _{H\in \mathcal {P}(n)}(-1)^{|K\cap H|}f(x^{c,H})]. \end{aligned}$$
(13)

In order to simplify the notation, we set \(D_\bullet :=D_{\{1,\ldots ,n\}}\) and \(\mathbb {R}_\bullet :=\mathbb {R}_{\{1,\ldots ,n\}}\). Consequently,

$$\begin{aligned}&D_\bullet \textstyle =\bigcap _{k=1}^n\{(z_1,\ldots ,z_n)\in D:z_k\not \in \mathbb {R}\}\ne \emptyset , \end{aligned}$$
(14)
$$\begin{aligned}&\mathbb {R}_\bullet \textstyle =\bigcup _{k=1}^n\{(x_1,\ldots ,x_n)\in A^n:x_k\in \mathbb {R}\}, \end{aligned}$$
(15)
$$\begin{aligned}&\Omega _{D_\bullet }\textstyle =\Omega _D\setminus \mathbb {R}_\bullet =\bigcap _{k=1}^n\{(x_1,\ldots ,x_n)\in \Omega _D:{\text {Im}}(x_k)\ne 0\}\ne \emptyset . \end{aligned}$$
(16)

Moreover, we set

$$\begin{aligned} \mathbb {R}_\emptyset :=\emptyset . \end{aligned}$$
(17)

Remark 2.20

According to Definitions 2.18 and 2.19, and (17), we can also say that the spherical value of f is the spherical \(\emptyset \)-derivative of f, that is \(f'_{s,\emptyset }:={f}_s^\circ \). \(\square \)

Proposition 2.21

Let \(f:\Omega _D\rightarrow A\) be a slice function. The following assertions hold.

  • \((\mathrm {i})\) For each \(x\in \Omega _D\), the spherical value \({f}_s^\circ \) is constant on \({\mathbb {S}}_x\). For each \(K\in \mathcal {P}(n)\setminus \{\emptyset \}\) and for each \(x\in \Omega _D\setminus \mathbb {R}_K\), the spherical K-derivative \(f'_{s,K}\) is constant on \({\mathbb {S}}_x\). More precisely, if \(f=\mathcal {I}(F)\), then

    $$\begin{aligned} {f}_s^\circ (x)=F_\emptyset (z)\,\text { for all }x\in \Omega _D \end{aligned}$$
    (18)

    and

    $$\begin{aligned} f'_{s,K}(x)=\beta _K^{-1}F_K(z)\,\hbox { for all }x\in \Omega _D\setminus \mathbb {R}_K, \end{aligned}$$
    (19)

    where \(z:=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\in D\) if \(x=(\alpha _1+J_1\beta _1,\ldots ,\alpha _n+J_n\beta _n)\in \Omega _D\).

  • \((\mathrm {ii})\) If \(x=(x_1,\ldots ,x_n)\in \Omega _D\setminus \mathbb {R}_L\) for some \(L\in \mathcal {P}(n)\) and \(x_h\in \mathbb {R}\) for all \(h\in \{1,\ldots ,n\}\setminus L\), then

    $$\begin{aligned} \textstyle f(x)={f}_s^\circ (x)+\sum _{K\in \mathcal {P}(n)\setminus \{\emptyset \},K\subset L}[{\text {Im}}_K(x),f'_{s,K}(x)]. \end{aligned}$$
    (20)

    In particular, for all \(x\in \Omega _D\setminus \mathbb {R}_\bullet \), we have:

    $$\begin{aligned} \textstyle f(x)={f}_s^\circ (x)+\sum _{K\in \mathcal {P}(n)\setminus \{\emptyset \}}[{\text {Im}}_K(x),f'_{s,K}(x)]. \end{aligned}$$
    (21)
  • \((\mathrm {iii})\) If \(x=(x_1,\ldots ,x_n)\in \Omega _D\setminus \mathbb {R}_L\) for some \(L\in \mathcal {P}(n)\) and \(x_h\in \mathbb {R}\) for all \(h\in \{1,\ldots ,n\}\setminus L\), then f is constant on \({\mathbb {S}}_x\) if and only if \(f'_{s,K}(x)=0\) for all \(K\in \mathcal {P}(n)\setminus \{\emptyset \}\) with \(K\subset L\). In this case, f takes the value \({f}_s^\circ (x)\) on \({\mathbb {S}}_x\).

    In particular, for each \(x\in \Omega _D\setminus \mathbb {R}_\bullet \), f is constant on \({\mathbb {S}}_x\) if and only if \(f'_{s,K}(x)=0\) for all \(K\in \mathcal {P}(n)\setminus \{\emptyset \}\).

Proof

Point \((\mathrm {i})\) follows immediately from the fact that the stem functions inducing \({f}_s^\circ \) and the \(f'_{s,K}\)’s are A-valued.

Let \(x=(\alpha _1+J_1\beta _1,\ldots ,\alpha _n+J_n\beta _n)\in \Omega _D\setminus \mathbb {R}_L\) for some \(L\in \mathcal {P}(n)\) and \(\beta _h=0\) for all \(h\in \{1,\ldots ,n\}\setminus L\), and let \(z=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\in D\). Denote \(F=\sum _{K\in \mathcal {P}(n)}e_KF_k\) the stem function inducing f. By (3), if \(K\in \mathcal {P}(n)\) with \(K\not \subset L\) , then \(F_K(z)=0\). Consequently, \(f(x)=\sum _{K\in \mathcal {P}(n),K\subset L}[J_K,F_K(z)]\), where \(J=(J_1,\ldots ,J_n)\). On the other hand, by the very definitions of spherical value and derivatives, we deduce:

$$\begin{aligned} f(x)&\textstyle =\sum _{K\in \mathcal {P}(n),K\subset L}[J_K,F_K(z)]\\&={f}_s^\circ (x)+\sum _{K\in \mathcal {P}(n)\setminus \{\emptyset \},K\subset L}[{\text {Im}}_K(x),\beta _K^{-1}F_K(z)]\\&\textstyle ={f}_s^\circ (x)+\sum _{K\in \mathcal {P}(n)\setminus \{\emptyset \},K\subset L}[{\text {Im}}_K(x),f'_{s,K}(x)]. \end{aligned}$$

This proves (20), which reduces to (21) when \(L=\{1,\ldots ,n\}\).

Let us prove \((\mathrm {iii})\). If \(f'_{s,K}(x)=0\) for each \(K\in \mathcal {P}(n)\setminus \{\emptyset \}\) with \(K\subset L\), then \((\mathrm {i})\) and (20) imply at once that such \(f'_{s,K}\)’s vanish on the whole \({\mathbb {S}}_x\) and f is constantly equal to \({f}_s^\circ (x)\) on \({\mathbb {S}}_x\). Finally, suppose that f is constantly equal on \({\mathbb {S}}_x\) to some \(a\in A\). Choose \(K\in \mathcal {P}(n)\setminus \{\emptyset \}\) with \(K\subset L\). Since \(x\in \Omega _D\setminus \mathbb {R}_K\) and \(x^{c,H}\in {\mathbb {S}}_x\) for all \(H\in \mathcal {P}(n)\), (12) and Lemma 2.11 ensure that

$$\begin{aligned} f'_{s,K}(x)&\textstyle =2^{-n}[({\text {Im}}_K(x))^{-1},\sum _{H\in \mathcal {P}(n)}(-1)^{|K\cap H|}a]\\&\textstyle =2^{-n}\big (\sum _{H\in \mathcal {P}(n)}(-1)^{|K\cap H|}\big )[({\text {Im}}_K(x))^{-1},a]\\&=\delta _{K,\emptyset }[({\text {Im}}_K(x))^{-1},a]=0. \end{aligned}$$

The proof is complete. \(\square \)

We now show that there exists a relation between the spherical value and derivatives of f and their one-variable analogues introduced in [9,  Definition 6]. Let \(z=(z_1,\ldots ,z_n)\in D\) and let \(h\in \{1,\ldots ,n\}\). Denote \(D_h(z)\) the subset of \(\mathbb {C}\) defined by

$$\begin{aligned} D_h(z):=\{w\in \mathbb {C}:(z_1,\ldots ,z_{h-1},w,z_{h+1},\ldots ,z_n)\in D\}. \end{aligned}$$
(22)

Since D is invariant under all the complex conjugations of \(\mathbb {C}^n\), it follows immediately that \(D_h(z)\) is invariant under the complex conjugation of \(\mathbb {C}\). Note that \(D_h(z)\ne \emptyset \), because it contains \(z_h\). Moreover, \(D_h(z)\) is open in \(\mathbb {C}\) if D is open in \(\mathbb {C}^n\).

Let \(x=(x_1,\ldots ,x_n)\in \Omega _D\). Denote \(\Omega _{D,h}(x)\) the subset of \(Q_A\) defined by

$$\begin{aligned} \Omega _{D,h}(x):=\{a\in A:(x_1,\ldots ,x_{h-1},a,x_{h+1},\ldots ,x_n)\in \Omega _D\}. \end{aligned}$$
(23)

Suppose that \(x\in \Omega _{\{z\}}\). Let us show that \(\Omega _{D_h(z)}=\Omega _{D,h}(x)\). First, note that, if we write \(z=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\) with \(\alpha _1,\ldots ,\alpha _n,\beta _1,\ldots ,\beta _n\in \mathbb {R}\), then \(x_\ell =\alpha _\ell +J_\ell \beta _\ell \) for each \(\ell \in \{1,\ldots ,n\}\) and for some \(J_\ell \in {\mathbb {S}}_A\). Let \(a\in \Omega _{D_h(z)}\). Write \(a=\alpha +J\beta \) with \(\alpha ,\beta \in \mathbb {R}\) and \(J\in {\mathbb {S}}_A\). By definition of \(\Omega _{D_h(z)}\) and \(D_h(z)\), we have that \(\alpha +i\beta \in D_h(z)\) and

$$\begin{aligned} (\alpha _1+i\beta _1,\ldots ,\alpha _{h-1}+i\beta _{h-1},\alpha +i\beta ,\alpha _{h+1}+i\beta _{h+1},\ldots ,\alpha _n+i\beta _n)\in D, \end{aligned}$$

respectively. The definitions of \(\Omega _D\) and \(\Omega _{D,h}(x)\) imply that \((x_1,\ldots ,x_{h-1},a,x_{h+1},\ldots ,x_n)\in \Omega _D\) and \(a\in \Omega _{D,h}(x)\), respectively. Vice versa, if \(a\in A\) with \((x_1,\ldots ,x_{h-1},a,x_{h+1},\ldots ,x_n)\in \Omega _D\), then there exists \(z'=(z'_1,\ldots ,z'_n)\in D\) with \(z'_\ell =\alpha '_\ell +i\beta '_\ell \) and \(J'_\ell \in {\mathbb {S}}_A\) for each \(\ell \in \{1,\ldots ,n\}\) such that \(\alpha '_h+J'_h\beta '_h=a\), and \(\alpha _\ell +J_\ell \beta _\ell =x_\ell =\alpha '_\ell +J'_\ell \beta '_\ell \) for each \(\ell \in \{1,\ldots ,n\}\setminus \{h\}\). Let \(\ell \ne h\). Note that, if \(\beta _\ell =0\), then \(\beta '_\ell =0\) as well, and \(\alpha _\ell =x_\ell =\alpha '_h\). If \(\beta _\ell \ne 0\), then either \((\beta '_\ell ,J'_\ell )=(\beta _\ell ,J_\ell )\) or \((\beta '_\ell ,J'_\ell )=(-\beta _\ell ,-J_\ell )\). Define \(H:=\{\ell \in \{1,\ldots ,n\}\setminus \{h\}:\beta _\ell \ne 0,(\beta '_\ell ,J'_\ell )=(-\beta _\ell ,-J_\ell )\}\). Since D is invariant under all complex conjugations of \(\mathbb {C}^n\), it follows that

$$\begin{aligned} (z_1,\ldots ,z_{h-1},\alpha '_h+i\beta '_h,z_{h+1},\ldots ,z_n)= \overline{\,z'\,}^{H}\in D. \end{aligned}$$

Hence \(a=\alpha '_h+J'_h\beta '_h\in \Omega _{D_h(z)}\). We have just proven that

$$\begin{aligned} \Omega _{D_h(z)}=\Omega _{D,h}(x) \end{aligned}$$
(24)

for all \(z=(z_1,\ldots ,z_n)\in D\), \(x=(x_1,\ldots ,x_n)\in \Omega _{\{z\}}\subset \Omega _D\) and \(h\in \{1,\ldots ,n\}\).

Definition 2.22

Let \(g:\Omega _D\rightarrow A\) be a function and let \(h\in \{1,\ldots ,n\}\). We say that g is a slice function w.r.t. \(\mathrm {x}_h\) if, for each \(y=(y_1,\ldots ,y_n)\in \Omega _D\), the restriction function \(g_h^{\scriptscriptstyle (y)}:\Omega _{D,h}(y)\rightarrow A\), defined by

$$\begin{aligned} g_h^{\scriptscriptstyle (y)}(x_h):=g(y_1,\ldots ,y_{h-1},x_h,y_{h+1},\ldots ,y_n), \end{aligned}$$

is a slice function. \(\square \)

Let \(g:\Omega _D\rightarrow A\) be a slice function w.r.t. \(\mathrm {x}_h\), let \(y\in \Omega _D\), let \({(g_h^{\scriptscriptstyle (y)})}_s^\circ :\Omega _{D,h}(y)\rightarrow A\) and \((g_h^{\scriptscriptstyle (y)})'_s:\Omega _{D,h}(y)\setminus \mathbb {R}\rightarrow A\) be the usual spherical value and spherical derivative of the one variable slice function \(g_h^{\scriptscriptstyle (y)}\), respectively. If z is a point of D such that \(y\in \Omega _{\{z\}}\) and \(g_h^{\scriptscriptstyle (y)}\) is induced by the stem function \(G_1+iG_2:D_h(z)\rightarrow A\otimes \mathbb {R}^2\), then \({(g_h^{\scriptscriptstyle (y)})}_s^\circ (x_h)=G_1(w)\) for all \(x_h=\alpha _h+J_h\beta _h\in \Omega _{D_h(z)}\), where \(w:=\alpha _h+i\beta _h\in D_h(z)\), and \((g_h^{\scriptscriptstyle (y)})'_s(x_h)=\beta _h^{-1} G_2(w)\) if \(\beta _h\ne 0\). As a consequence, we have that \({(g_h^{\scriptscriptstyle (y)})}_s^\circ (x_h)=\frac{1}{2}(g_h^{\scriptscriptstyle (y)}(x_h)+g_h^{\scriptscriptstyle (y)}((x_h)^c))\) and \((g_h^{\scriptscriptstyle (y)})'_s(x_h)=\frac{1}{2}({\text {Im}}(x_h))^{-1}(g_h^{\scriptscriptstyle (y)}(x_h)-g_h^{\scriptscriptstyle (y)}((x_h)^c))\).

Assume that g is a slice function w.r.t. \(\mathrm {x}_h\). Then, for each \(e\in \{0,1\}\), we define the function \(\mathcal {D}_{\mathrm {x}_h}^0g:\Omega _D\rightarrow A\) and \(\mathcal {D}_{\mathrm {x}_h}^1g:\Omega _D\setminus \mathbb {R}_{\{h\}}\rightarrow A\) by setting

$$\begin{aligned} \mathcal {D}_{\mathrm {x}_h}^0g(x):={(g_h^{\scriptscriptstyle (x)})}_s^\circ (x_h)\, \text { for all }x=(x_1,\ldots ,x_n)\in \Omega _D \end{aligned}$$
(25)

and

$$\begin{aligned} \mathcal {D}_{\mathrm {x}_h}^1g(x):=(g_h^{\scriptscriptstyle (x)})'_s(x_h)\, \text { for all } x=(x_1,\ldots ,x_n)\in \Omega _D\setminus \mathbb {R}_{\{h\}}. \end{aligned}$$
(26)

Given any \(K\in \mathcal {P}(n)\) and \(h\in \{1,\ldots ,n\}\), define \(K_h:=K\cap \{1,\ldots ,h\}\).

Proposition 2.23

Assume that \(n\ge 2\). Let \(f:\Omega _D\rightarrow A\) be a slice function, let \(K\in \mathcal {P}(n)\) and let \(\epsilon :\{1,\ldots ,n\}\rightarrow \{0,1\}\) be the characteristic function of K. Then f is a slice function w.r.t. \(\mathrm {x}_1\) and, for each \(h\in \{2,\ldots ,n\}\), the function \(\mathcal {D}^{\epsilon (h-1)}_{\mathrm {x}_{h-1}}\cdots \mathcal {D}^{\epsilon (1)}_{\mathrm {x}_1}f:\Omega _D\setminus \mathbb {R}_{K_{h-1}}\rightarrow A\), obtained iterating (25) and (26) as follows \(\mathcal {D}^{\epsilon (h-1)}_{\mathrm {x}_{h-1}}\cdots \mathcal {D}^{\epsilon (1)}_{\mathrm {x}_1}f:=\mathcal {D}^{\epsilon (h-1)}_{\mathrm {x}_{h-1}}(\mathcal {D}^{\epsilon (h-2)}_{\mathrm {x}_{h-2}}\cdots (\mathcal {D}^{\epsilon (2)}_{\mathrm {x}_2}(\mathcal {D}^{\epsilon (1)}_{\mathrm {x}_1}f))\cdots )\), is a well-defined slice function w.r.t. \(\mathrm {x}_h\). Moreover, it holds:

$$\begin{aligned} \mathcal {D}^{\epsilon (n)}_{\mathrm {x}_n}\cdots \mathcal {D}^{\epsilon (1)}_{\mathrm {x}_1}f:=\mathcal {D}^{\epsilon (n)}_{\mathrm {x}_n}(\mathcal {D}^{\epsilon (n-1)}_{\mathrm {x}_{n-1}}\cdots \mathcal {D}^{\epsilon (1)}_{\mathrm {x}_1}f)= {\left\{ \begin{array}{ll} {f}_s^\circ &{}{} \text{ if } K=\emptyset ,\\ f'_{s,K}&{}{} \text{ if } K\ne \emptyset . \end{array}\right. } \end{aligned}$$

Proof

Let \(F=\sum _{H\in \mathcal {P}(n)}e_HF_H:D\rightarrow A\otimes \mathbb {R}^{2^n}\) be the stem function inducing f, let \(y=(y_1,\ldots ,y_n)=(\alpha _1+I_1\beta _1,\ldots ,\alpha _n+I_n\beta _n)\in \Omega _D\), let \(I:=(I_1,\ldots ,I_n)\) and let \(w=(w_1,\ldots ,w_n):=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\in D\). Let us prove by induction on \(h\in \{1,\ldots ,n\}\) that the two following properties hold true:

  1. (a)

    \(\mathcal {D}^{\epsilon (h-1)}_{{\mathrm {x}_{h-1}}}\cdots \mathcal {D}^{\epsilon (1)}_{\mathrm {x}_1}f\) is a slice function w.r.t. \(\mathrm {x}_h\), where for convention \(\mathcal {D}^{\epsilon (h-1)}_{\mathrm {x}_{h-1}}\cdots \mathcal {D}^{\epsilon (1)}_{\mathrm {x}_1}f:=f\) if \(h=1\).

  2. (b)

    \(\mathcal {D}^{\epsilon (h)}_{\mathrm {x}_h}\cdots \mathcal {D}^{\epsilon (1)}_{\mathrm {x}_1}f(y)= (\beta _{K_h})^{-1}\sum _{H\in \mathcal {P}(n),H_h=\emptyset }[I_H,F_{H\cup K_h}(z)]\), where \(\beta _\emptyset :=1\).

First, we consider the case \(h=1\). It holds:

$$\begin{aligned} \textstyle f(x_1,y')=\sum _{H\in \mathcal {P}(n),1\not \in H}[J_H,F_H(z_1,w')]+J_1\big (\sum _{H\in \mathcal {P}(n),1\not \in H}[J_H,F_{H\cup \{1\}}(z_1,w')]\big ), \end{aligned}$$
(27)

where \(x_1=\alpha _1+J_1\beta _1\in \Omega _{D,1}(y)\), \(y':=(y_2,\ldots ,y_n)\), \(z_1:=\alpha _1+i\beta _1\in D_1(w)\), \(w':=(w_2,\ldots ,w_n)\) and \(J=(J_1,I_2,\ldots ,I_n)\). Define the functions \(F_1,F_2:D_1(w)\rightarrow A\) by setting

$$\begin{aligned} F_1(z_1)&\textstyle :=\sum _{H\in \mathcal {P}(n),1\not \in H}[J_H,F_H(z_1,w')],\\ F_2(z_1)&\textstyle :=\sum _{H\in \mathcal {P}(n),1\not \in H}[J_H,F_{H\cup \{1\}}(z_1,w')]. \end{aligned}$$

It is immediate to verify that \(F_1+iF_2:D_1(w)\rightarrow A\otimes \mathbb {R}^2\) is a stem function. Consequently, f is slice w.r.t. \(\mathrm {x}_1\). Moreover, we have:

$$\begin{aligned} \textstyle \mathcal {D}^0_{\mathrm {x}_1}f(y)=\sum _{H\in \mathcal {P}(n),1\not \in H}[I_H,F_H(z)] \end{aligned}$$

and

$$\begin{aligned} \textstyle \mathcal {D}^1_{\mathrm {x}_1}f(y)=\beta _1^{-1}\sum _{H\in \mathcal {P}(n),1\not \in H}[I_H,F_{H\cup \{1\}}(z)]. \end{aligned}$$

This proves that f satisfies (a) and (b) for \(h=1\).

Assume (a) and (b) are verified for some \(h\in \{1,\ldots ,n-1\}\). By (b), we deduce:

$$\begin{aligned}&\mathcal {D}^{\epsilon (h)}_{\mathrm {x}_h}\cdots \mathcal {D}^{\epsilon (1)}_{\mathrm {x}_1}f(y'',x_{h+1},{\hat{y}})\nonumber \\&\quad =\textstyle (\beta _{K_{h-1}}\beta _{h,K})^{-1}\sum _{H\in \mathcal {P}(n),H_{h+1}=\emptyset }[L_H,F_{H\cup K_h}(z'',z_{h+1},{\hat{z}})]\nonumber \\&\qquad \textstyle +J_{h+1} \big ((\beta _{K_{h-1}}\beta _{h,K})^{-1}\sum _{H\in \mathcal {P}(n),H_{h+1}=\emptyset }[L_H, F_{H\cup K_h\cup \{h+1\}}(z'',z_{h+1},{\hat{z}})]\big ), \end{aligned}$$
(28)

where \(x_h=\alpha _h+J_h\beta _h\in \Omega _{D,h}(y)\), \(\beta _{h,K}=\beta _h\) if \(h\in K\), \(\beta _{h,K}=1\) if \(h\not \in K\), \(y''=(y_1,\ldots ,y_{h})\), \({\hat{y}}=(y_{h+2},\ldots ,y_n)\), \(z_h=\alpha _h+i\beta _h\in D_h(z)\), \(z''=(z_1,\ldots ,z_{h})\), \({\hat{z}}=(z_{h+2},\ldots ,z_n)\) and \(L=(I_1,\ldots ,I_{h},J_{h+1},I_{h+2},\ldots ,I_n)\). Here \({\hat{y}}\) and \({\hat{z}}\) are omitted if \(h+1=n\). Proceeding as above, it is immediate to verify that \(\mathcal {D}^{\varepsilon (h)}_{\mathrm {x}_h}\cdots \mathcal {D}^{\varepsilon (1)}_{\mathrm {x}_1}f\) is slice w.r.t. \(\mathrm {x}_{h+1}\), i.e. (a) is satisfied for \(h+1\). Moreover, we have:

$$\begin{aligned} \mathcal {D}^0_{\mathrm {x}_{h+1}}\mathcal {D}^{\epsilon (h)}_{\mathrm {x}_h}\cdots \mathcal {D}^{\epsilon (1)}_{\mathrm {x}_1}f(y)&\textstyle =(\beta _{K_h})^{-1}\sum _{H\in \mathcal {P}(n),H_{h+1}=\emptyset }[I_H, F_{H\cup K_h}(z)],\\ \mathcal {D}^1_{\mathrm {x}_{h+1}}\mathcal {D}^{\epsilon (h)}_{\mathrm {x}_h}\cdots \mathcal {D}^{\epsilon (1)}_{\mathrm {x}_1}f(y)&\textstyle = \beta _{h+1}^{-1}\big (\beta _{K_h}\big )^{-1}\sum _{H\in \mathcal {P}(n),H_{h+1}=\emptyset }[I_H,F_{H\cup K_h\cup \{h+1\}}(z)]. \end{aligned}$$

In both cases, the last two expressions are equal to

$$\begin{aligned} \textstyle (\beta _{K_{h+1}})^{-1}\sum _{H\in \mathcal {P}(n),H_{h+1}=\emptyset }[I_H, F_{H\cup K_{h+1}}(z)] \end{aligned}$$

and the induction step works. When \(h=n\), the right-hand side of (b) becomes \(F_\emptyset (z)\) if \(K=\emptyset \), and \(\beta _K^{-1}F_K(z)\) if \(K\ne \emptyset \). This completes the proof. \(\square \)

Definition 2.24

Assume that \(n\ge 2\). Let \(h\in \{2,\ldots ,n\}\) and let \(\epsilon :\{1,\ldots ,h-1\}\rightarrow \{0,1\}\) be any function. Given a slice function \(f:\Omega _D\rightarrow A\), we define the truncated spherical \(\epsilon \)-derivative \(\mathcal {D}_\epsilon f:\Omega _D\setminus \mathbb {R}_{\epsilon ^{-1}(1)}\rightarrow A\) of f by setting \(\mathcal {D}_\epsilon f:=\mathcal {D}^{\epsilon (h-1)}_{\mathrm {x}_{h-1}}\cdots \mathcal {D}^{\epsilon (1)}_{\mathrm {x}_1}f\), and we say that such a derivative has order \(h-1\). For convention, we define also the truncated spherical \(\emptyset \)-derivative \(\mathcal {D}_\emptyset f:\Omega _D\rightarrow A\) of f by setting \(\mathcal {D}_\emptyset f:=f\), and we say that such a derivative has order 0. \(\square \)

Proposition 2.23 asserts that each h-order truncated spherical derivative of a slice function is a well-defined slice function w.r.t. \(\mathrm {x}_h\). Moreover, a by-product of the proof of the mentioned proposition reads as follows. If \(F:D\rightarrow A\otimes \mathbb {R}^{2^n}\) is the stem function inducing f, then

$$\begin{aligned} \textstyle \mathcal {D}_\epsilon f(x)=(\beta _{\epsilon ^{-1}(1)})^{-1}\sum _{H\in \mathcal {P}(n),H_{h-1}=\emptyset }[J_H,F_{H\cup \epsilon ^{-1}(1)}(z)] \end{aligned}$$
(29)

for all \(x=(\alpha _1+J_1\beta _1,\ldots ,\alpha _n+J_n\beta _n)\in \Omega _D\setminus \mathbb {R}_{\epsilon ^{-1}(1)}\) with \(z:=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\) and \(J:=(J_1,\ldots ,J_n)\), where \(\beta _\emptyset :=1\). Formula (29) shows that every truncated spherical derivative \(\mathcal {D}_\epsilon f\) is a slice function of n variables on \(\Omega _D\setminus \mathbb {R}_{\epsilon ^{-1}(1)}\), induced by the stem function

$$\begin{aligned} \textstyle G(z)=\sum _{H\in \mathcal {P}(n),H_{h-1}=\emptyset }e_H (\beta _{\epsilon ^{-1}(1)})^{-1}F_{H\cup \epsilon ^{-1}(1)}(z). \end{aligned}$$

Smoothness

By Assumption 2.1, the real alternative \(^*\)-algebra A we are working with has finite dimension and, as a finite dimensional real vector space, A is equipped with the natural \(\mathscr {C}^\omega \) manifold structure defined by the global coordinate systems associated with its real vector bases. Here ‘\(\,\mathscr {C}^\omega \,\)’ means ‘real analytic’. For each \(n\ge 1\), we equip \(A^n\) with the corresponding product structure of \(\mathscr {C}^\omega \) manifolds. We call the underlying topology on \(A^n\) as Euclidean topology of \(A^n\). Given any non-empty subset S of \(A^n\), we equip S with the relative topology induced by the Euclidean one of \(A^n\). We call such a topology on S as Euclidean topology of S. If in addition S is open in \(A^n\), then we always assume that S is equipped with the \(\mathscr {C}^\omega \) manifold structure induced by the one of \(A^n\).

Similarly, we equip D with the Euclidean topology induced by the one of \(\mathbb {C}=\mathbb {R}^2\) and, in the case D is open in \(\mathbb {C}\), we always assume that D is equipped with the \(\mathscr {C}^\omega \) manifold structure induced by the one of \(\mathbb {C}=\mathbb {R}^2\).

As usual, given two topological spaces X and Y, we denote \(\mathscr {C}^0(X,Y)\) the set of all continuous maps from X to Y. If \(r\in (\mathbb {N}\setminus \{0\})\cup \{\infty ,\omega \}\) and X and Y are equipped with some \(\mathscr {C}^r\) manifold structures (for instance, \(\mathscr {C}^\omega \) manifold structures), then the symbol \(\mathscr {C}^r(X,Y)\) indicates the set of all \(\mathscr {C}^r\) maps from X to Y. In the latter case, given any non-empty subset S of X and a map \(f:S\rightarrow Y\), we say that f is a \(\mathscr {C}^r\) map if there exist an open neighborhood U of S in X and a map \(g:U\rightarrow Y\) such that \(g(x)=f(x)\) for all \(x\in S\) and, equipping U with the natural \(\mathscr {C}^r\) manifold structure induced by the one of X, g belongs to \(\mathscr {C}^r(U,Y)\). We denote \(\mathscr {C}^r(S,Y)\) the set of all \(\mathscr {C}^r\) maps from S to Y.

Definition 2.25

We define

  • \(\mathrm {Stem}^0(D,A\otimes \mathbb {R}^{2^n})\) as the set of all continuous stem functions from D to \(A\otimes \mathbb {R}^{2^n}\), i.e. the set of all stem functions \(F=\sum _{K\in \mathcal {P}(n)}e_KF_K:D\rightarrow A\otimes \mathbb {R}^{2^n}\) such that each \(F_K\) belongs to \(\mathscr {C}^0(D,A)\),

  • \(\mathcal {S}^0(\Omega _D,A)\) as the set of slice functions from \(\Omega _D\) to A induced by continuous stem functions, i.e. \(\mathcal {S}^0(\Omega _D,A):=\mathcal {I}(\mathrm {Stem}^0(D,A\otimes \mathbb {R}^{2^n}))\),

and, in the case D is open in \(\mathbb {C}\) and \(r\in (\mathbb {N}\setminus \{0\})\cup \{\infty ,\omega \}\),

  • \(\mathrm {Stem}^r(D,A\otimes \mathbb {R}^{2^n})\) as the set of all \(\mathscr {C}^r\) stem functions from D to \(A\otimes \mathbb {R}^{2^n}\), i.e. the set of all stem functions \(F=\sum _{K\in \mathcal {P}(n)}e_KF_K:D\rightarrow A\otimes \mathbb {R}^{2^n}\) such that each \(F_K\) belongs to \(\mathscr {C}^r(D,A)\),

  • \(\mathcal {S}^r(\Omega _D,A)\) as the set of slice functions from \(\Omega _D\) to A induced by \(\mathscr {C}^r\) stem functions, i.e. \(\mathcal {S}^r(\Omega _D,A):=\mathcal {I}(\mathrm {Stem}^r(D,A\otimes \mathbb {R}^{2^n}))\). \(\square \)

Equip \(\mathbb {N}\cup \{\infty ,\omega \}\) with the unique total ordering \(\le \), extending the one \(\le \) of \(\mathbb {N}\), by requiring that \(s\le \infty \) for all \(s\in \mathbb {N}\), and \(\infty \le \omega \). Denote \(\lfloor s\rfloor \) the integer part of \(s\in \mathbb {R}\).

Theorem 2.26

The following assertions hold.

  • \((\mathrm {i})\) If \({\mathbb {S}}_A\) is compact, then \(\mathcal {S}^0(\Omega _D,A)\subset \mathscr {C}^0(\Omega _D,A)\).

  • \((\mathrm {ii})\) Suppose that D is open in \(\mathbb {C}^n\). Let \(r\in \mathbb {N}\cup \{\infty ,\omega \}\) such that \(r\ge 2^n-1\), and let \(\mathbf {w}_n(r)\) be the element of \(\mathbb {N}\cup \{\infty ,\omega \}\) defined by \(\mathbf {w}_n(r):=r\) if \(r\in \{\infty ,\omega \}\) and

    $$\begin{aligned} \textstyle \mathbf {w}_n(r):=\left\lfloor \frac{r-2^n+1}{2^n}\right\rfloor =\left\lfloor \frac{r+1}{2^n}\right\rfloor -1 \end{aligned}$$

    if \(r\in \mathbb {N}\) and \(r\ge 2^n-1\). Then it holds:

    $$\begin{aligned} \mathcal {S}^r(\Omega _D,A)\subset \mathscr {C}^{\mathbf {w}_n(r)}(\Omega _D,A). \end{aligned}$$

    In particular, we have \(\mathcal {S}^\infty (\Omega _D,A)\subset \mathscr {C}^\infty (\Omega _D,A)\) and \(\mathcal {S}^\omega (\Omega _D,A)\subset \mathscr {C}^\omega (\Omega _D,A)\).

Proof

Choose a real vector basis \(\mathcal {B}=(u_1,\ldots ,u_d)\) of A with \(u_1=1\), and denote \(\pi _\mathbb {R}:A\rightarrow \mathbb {R}\) the projection of A onto the first component of the coordinates induced by \(\mathcal {B}\), i.e. the real linear function sending each \(a=\sum _{h=1}^da_hu_h\in A\) into \(a_1\in \mathbb {R}\). Define the functions \(\theta ,\eta ,\xi :A\rightarrow \mathbb {R}\) and \(v,w:A\rightarrow \mathbb {C}\) by setting

$$\begin{aligned} \theta (a):=\pi _\mathbb {R}({\text {Re}}(a)), \quad \eta (a):=\pi _\mathbb {R}(n({\text {Im}}(x))), \quad \xi (a):=\sqrt{|\eta (a)|} \end{aligned}$$

and

$$\begin{aligned} v(a):=\theta (a)+i\xi (a), \quad w(a):=\theta (a)+i\eta (a). \end{aligned}$$

Note that \(\theta ,\eta \in \mathscr {C}^\omega (A,\mathbb {R})\), \(\xi \in \mathscr {C}^0(A,\mathbb {R})\), \(v\in \mathscr {C}^0(A,\mathbb {C})\) and \(w\in \mathscr {C}^\omega (A,\mathbb {C})\). Moreover, it holds

$$\begin{aligned}&v(\alpha +J\beta )=\alpha +i|\beta |, \end{aligned}$$
(30)
$$\begin{aligned}&w(\alpha +J\beta )=\alpha +i\beta ^2 \end{aligned}$$
(31)

for all \(\alpha ,\beta \in \mathbb {R}\) and \(J\in {\mathbb {S}}_A\). Define also the maps \(v_n,w_n:A^n\rightarrow \mathbb {C}^n\) by setting

$$\begin{aligned} v_n(x_1,\ldots ,x_n):=(v(x_1),\ldots ,v(x_n)) \,\text { and }\, w_n(x_1,\ldots ,x_n):=(w(x_1),\ldots ,w(x_n)). \end{aligned}$$

Let C be the closed subset of A defined by \(C:=\xi ^{-1}(0)=\eta ^{-1}(0)\). Since \(C\cap Q_A=\mathbb {R}\), we have that \((A\setminus C)\cap Q_A=Q_A\setminus C=Q_A\setminus \mathbb {R}\), and hence \((A\setminus C)^n\cap \Omega _D=\Omega _D\setminus \mathbb {R}_\bullet \). Let \(\mathrm {j}:A\setminus C\rightarrow A\) be the continuous map defined by

$$\begin{aligned} \textstyle \mathrm {j}(a):=\frac{1}{\xi (a)}{\text {Im}}(a). \end{aligned}$$

Define \(\mathrm {J}_\emptyset :=1\) and, for each \(K=\{k_1,\ldots ,k_p\}\in \mathcal {P}(n)\setminus \{\emptyset \}\) with \(k_1<\ldots <k_p\), define the continuous map \(\mathrm {J}_K:\Omega _D\setminus \mathbb {R}_{K}\rightarrow ({\mathbb {S}}_A)^{|K|}\) by

$$\begin{aligned} \mathrm {J}_K(x_1,\ldots ,x_n):=(\mathrm {j}(x_{k_1}),\ldots ,\mathrm {j}(x_{k_p})). \end{aligned}$$

Denote \(\mathrm {J}_\bullet :Q_A\setminus \mathbb {R}_\bullet \rightarrow ({\mathbb {S}}_A)^n\) the continuous map \(\mathrm {J}_{\{1,\ldots ,n\}}\). Note that \(\mathrm {J}_\bullet (\alpha _1+J_1|\beta _1|,\ldots ,\alpha _n+J_n|\beta _n|)=(J_1,\ldots ,J_n)\) if \((\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\in D_\bullet \) and \(J_1,\ldots ,J_n\in {\mathbb {S}}_A\).

Choose \(F=\sum _{K\in \mathcal {P}(n)}e_KF_K\in \mathrm {Stem}(D,A\otimes \mathbb {R}^{2^n})\) and define \(f:=\mathcal {I}(F)\).

Let us prove \((\mathrm {i})\).

Suppose that F is continuous and \({\mathbb {S}}_A\) is compact. For each \(H\in \mathcal {P}(n)\) and \(\ell \in \{0,1,\ldots ,n\}\), define \(Q(H):=\bigcap _{h\in \{1,\ldots ,n\}\setminus H}\{(x_1,\ldots ,x_n)\in \Omega _D:x_h\in \mathbb {R}\}\) and \(Q(\ell ):=\bigcup _{H\in \mathcal {P}(n),|H|\le \ell }Q(H)\). Note that the Q(H)’s and the \(Q(\ell )\)’s are closed subsets of \(\Omega _D\); moreover, \(Q(0)=Q(\emptyset )=\Omega _D\cap \mathbb {R}^n\) and \(Q(n)=Q(\{1,\ldots ,n\})=\Omega _D\). We will prove by induction on \(\ell \in \{0,1,\ldots ,n\}\) that the restriction \(f|_{Q(\ell )}\) of f to \(Q(\ell )\) is continuous. Evidently, if the latter assertion is true then f is continuous, because \(Q(n)=\Omega _D\). The case \(\ell =0\) follows immediately from the equality \(f(x)=F_\emptyset (v_n(x))\) for all \(x\in Q(0)=\Omega _D\cap \mathbb {R}^n\). Suppose the assertion is true for some \(\ell \in \{0,1,\ldots ,n-1\}\). Note that, for each \(H,L\in \mathcal {P}(n)\) with \(|H|=|L|=\ell +1\) and \(H\ne L\), we have that \(|H\cap L|\le \ell \) and hence

$$\begin{aligned} \textstyle Q(H)\cap Q(L)=\bigcap _{h\in \{1,\ldots ,n\}\setminus (H\cap L)}\{(x_1,\ldots ,x_n)\in \Omega _D:x_h\in \mathbb {R}\}\subset Q(\ell ). \end{aligned}$$

It follows that \(\{Q(H)\}_{H\in \mathcal {P}(n),|H|=\ell {+1}}\) is a finite closed cover of \(Q(\ell +1)\) and, for each \(H,L\in \mathcal {P}(n)\) with \(|H|=|L|=\ell +1\) and \(H\ne L\), the restrictions \(f|_{Q(H)}\) and \(f|_{Q(L)}\) are continuous on \(Q(H)\cap Q(L)\) by induction. Consequently, it suffices to show that, for each fixed \(H\in \mathcal {P}(n)\) with \(|H|=\ell +1\), \(f|_{Q(H)}\) is continuous. By induction, \(f|_{Q(\ell )}\) is continuous so the same is true for \(f|_P\), where \(P:=Q(H)\cap Q(\ell )\). The set P is closed in Q(H) and

$$\begin{aligned} Q(H)\setminus P=Q(H)\setminus Q(\ell )\subset \Omega _D\setminus \mathbb {R}_\bullet . \end{aligned}$$

Since \(f(x)=\sum _{K\in \mathcal {P}(n)}[\mathrm {J}_\bullet (x),F_K(v_n(x))]\) for all \(x\in \Omega _D\setminus \mathbb {R}_\bullet \), it follows that \(f|_{Q(H)\setminus P}\) is continuous. Now, in order to complete the proof of \((\mathrm {i})\), it suffices to show that, if \(\{y_m\}_{m\in \mathbb {N}}\) is a sequence in \(Q(H)\setminus P\) converging to some point \(x=(x_1,\ldots ,x_n)\in P\), then the sequence \(\{f(y_m)\}_{m\in \mathbb {N}}\) converges to f(x). Consider such a sequence \(\{y_m\}_{m\in \mathbb {N}}\) in \(Q(H)\setminus P\) and \(x=(x_1,\ldots ,x_n)\in P\). Define \(H^*:=\{h\in H:x_h\in \mathbb {R}\}\). Note that, by definition of P, \(H^*\ne \emptyset \). By the even-odd properties of the \(F_K\)’s, we deduce at once that

$$\begin{aligned} \textstyle f(y_m)=\sum _{K\in \mathcal {P}(n),K\subset H}[\mathrm {J}_K(y_m),F_K(v_n(y_m))] \end{aligned}$$
(32)

for all \(m\in \mathbb {N}\), and

$$\begin{aligned} \textstyle f(x)=\sum _{K\in \mathcal {P}(n),K\subset H\setminus H^*}[\mathrm {J}_K(x),F_K(v_n(x))]. \end{aligned}$$
(33)

Let \(K\in \mathcal {P}(n)\) with \(K\subset H\) and \(K\cap H^*\ne \emptyset \). Choose \(\nu \in K\cap H^*\) and observe that \(x_\nu \in \mathbb {R}\). Since \(v_n\) and \(F_K\) are continuous, the sequence \(\{F_K(v_n(y_m))\}_{m\in \mathbb {N}}\) converges to \(F_K(v_n(x))\). If we write \(v_n(x)=(z_1,\ldots ,z_n)\in \mathbb {C}^n\), then \(z_\nu =x_\nu \in \mathbb {R}\) and \(F_K(v_n(x))=0\). On the other hand, \(({\mathbb {S}}_A)^n\) is compact in \(A^n\) and hence it is bounded. It follows that the sequence \(\{[\mathrm {J}_K(y_m),F_K(v_n(y_m))]\}_{m\in \mathbb {N}}\) converges to zero. Consequently, by (32), \(\{f(y_m)\}_{m\in \mathbb {N}}\) converges to \(\sum _{K\in \mathcal {P}(n),K\subset H\setminus H^*}[\mathrm {J}_K(x),F_K(v_n(x))]\), which is equal to f(x) by (33).

It remains to show point \((\mathrm {ii})\).

Suppose that D is open in \(\mathbb {C}^n\). Assume that F is of class \(\mathscr {C}^r\) for \(r\in \mathbb {N}\) with \(r\ge 2^n-1\). Let \(\rho :\mathbb {Z}\rightarrow \mathbb {Z}\) be the function \(\rho (s):=\big \lfloor \frac{s-1}{2}\big \rfloor \) and, for each \(k\ge 1\), let \(\rho ^k:\mathbb {Z}\rightarrow \mathbb {Z}\) be the \(k^{\mathrm {th}}\)-iterated composition of \(\rho \) with itself. Since \(\rho \) is non-decreasing and \(r\ge 2^n\mathbf {w}_n(r)+2^n-1\), we have that \(\rho ^n\) is non-decreasing and

$$\begin{aligned} \rho ^n(r)\ge \mathbf {w}_n(r); \end{aligned}$$
(34)

indeed, it holds:

$$\begin{aligned} \rho ^n(r)&\ge \rho ^n(2^n\mathbf {w}_n(r)+2^n-1)\\&=\rho ^{n-1}(2^{n-1}\mathbf {w}_n(r)+2^{n-1}-1)=\ldots =\rho (2\mathbf {w}_n+1)=\mathbf {w}_n(r). \end{aligned}$$

Consider the component \(F_K:D\rightarrow A\) of F for some fixed \(K\in \mathcal {P}(n)\). If \(z=(z_1,\ldots ,z_n)=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\) are the coordinates of \(\mathbb {C}^n\), then \(F_K\) is even w.r.t. \(z_h\) if \(h\not \in K\) and it is odd w.r.t. \(z_h\) if \(h\in K\). By (34), \(\rho ^n(r)\) is non-negative, because \(\rho ^n(r)\ge \mathbf {w}_n(r)\ge 0\). Since \(F_K\) is of class \(\mathscr {C}^r\) and \(\rho ^n(r)\ge 0\), we can apply to \(F_K\) the representation results of Whitney for even-odd function along each variables \(z_1,\ldots ,z_n\), see [24] especially Remark at page 160. In this way, if \(W_n:\mathbb {C}^n\rightarrow \mathbb {C}^n\) is the \(\mathscr {C}^\omega \) map given by \(W_n(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n):=(\alpha _1+i\beta _1^2,\ldots ,\alpha _n+i\beta _n^2)\), then we obtain an open neighborhood U of \(W_n(D)\) in \(\mathbb {C}^n\) and a \(\mathscr {C}^{\rho ^n(r)}\) map \(F'_K:U\rightarrow A\) such that \(F_K(z)=\beta _KF'_K(W_n(z))\) for all \(z\in D\), where \(\beta _\emptyset :=1\). Using (34) again, we know that each function \(F'_k\) is also of class \(\mathscr {C}^{\mathbf {w}_n(r)}\). Note that \(w_n(\Omega _D)\subset W_n(D)\subset U\). Consequently, the set \(V:=(w_n)^{-1}(U)\) is an open neighborhood of \(\Omega _D\) in \(A^n\). Define the function \({\hat{f}}:V\rightarrow A\) by setting

$$\begin{aligned} \textstyle {\hat{f}}(x):=\sum _{K\in \mathcal {P}(n)}[{\text {Im}}_K(x),F'_K(w_n(x))]. \end{aligned}$$

The function \({\hat{f}}\) is of class \(\mathscr {C}^{\mathbf {w}_n(r)}\) and extends f to the whole V. As a consequence, f belongs to \(\mathscr {C}^{\mathbf {w}_n(r)}\), as desired. The proof in the case \(r\in \{\infty ,\omega \}\) is similar, but easier because the \(F'_K\)’s have the same \(\mathscr {C}^r\) regularity of \(F_K\). \(\square \)

Remark 2.27

(i) In the statement of point (i) of the preceding result, we cannot omit the compactness condition on \({\mathbb {S}}_A\), also in the one variable case. In Proposition 7(1) of [9] the compactness hypothesis is missing. Let A be the Clifford algebra \( C \ell _{1,1}={\mathbb {S}}{\mathbb {H}}\) of split-quaternions, equipped with the Clifford conjugation (see Sects. 3.2.1 and 3.2.2 of [17]). Given any element \(x=x_0+x_1e_1+x_2e_2+x_{12}e_{12}\) of \({\mathbb {S}}{\mathbb {H}}\) with \(x_0,x_1,x_2,x_{12}\in \mathbb {R}\), we have that \(t(x)=2x_0\) and \(n(x)=x_0^2-x_1^2+x_2^2-x_{12}^2\). It follows that \({\mathbb {S}}_A\) is the 2-hyperboloid of \(A\simeq \mathbb {R}^4\) given by the equations \(x_0=0=x_2^2-x_1^2-x_{12}^2-1\) and \(Q_A\) is the union of \(\mathbb {R}\) and the open cone \(x_2^2-x_1^2-x_{12}^2>0\). Note that \({\mathbb {S}}_A\) is not compact. Consider the continuous stem function \(F:\mathbb {C}\rightarrow A\otimes \mathbb {C}\) defined by \(F(\alpha +i\beta ):=i|\beta |^{\frac{1}{2}}\mathrm {sgn}(\beta )\), where \(\mathrm {sgn}(\beta )\) is equal to 1 if \(\beta >0\), \(-1\) if \(\beta <0\) and 0 if \(\beta =0\). If \(f:Q_A\rightarrow A\) is the one variable slice function induced by F, then \(f(x)=0\) for all \(x\in \mathbb {R}\), and \(f(x)=(x_2^2-x_1^2-x_{12}^2)^{-\frac{1}{4}}(x_1e_1+x_2e_2+x_{12}e_{12})\) for all \(x\in Q_A\setminus \mathbb {R}\). Let \(\alpha \in \mathbb {R}\) and, for each \(t>0\), let \(y_t\) be the point of \(Q_A\setminus \mathbb {R}\) defined by \(y_t:=\alpha +te_1+(t+t^4)e_2\). Since \(f(y_t)=t^{-\frac{1}{4}}(2+t^3)^{-\frac{1}{4}}(e_1+(1+t^3)e_2)\) for all \(t>0\), we have that \(\lim _{t\rightarrow 0^+}y_t=\alpha \) and \(\lim _{t\rightarrow 0^+}t^{\frac{1}{4}}f(y_t)=2^{-\frac{1}{4}}(e_1+e_2)\ne 0\). This proves that f is not continuous at \(\alpha \).

(ii) A by-product of the preceding proof is that, if D is open in \(\mathbb {C}^n\), then \(\mathcal {S}^r(\Omega _D,A)\subset \mathscr {C}^{\rho ^n(r)}(\Omega _D,A)\) for all \(r\in \mathbb {N}\) with \(\rho ^n(r)\ge 0\).

(iii) Thanks to the preceding proof, it is also quite evident that, if \(\Omega _D\cap \mathbb {R}_\bullet =\emptyset \), then \(\mathcal {S}^r(\Omega _D,A)\subset \mathscr {C}^r(\Omega _D,A)\) for all \(r\in \mathbb {N}\cup \{\infty ,\omega \}\). \(\square \)

Multiplicative structures on slice functions and polynomials

Let us introduce the concept of symmetric difference algebra, or \(\bigtriangleup \)-algebra for short.

Definition 2.28

Given a bilinear map \(\mathrm {b}:\mathbb {R}^{2^n}\times \mathbb {R}^{2^n}\rightarrow \mathbb {R}^{2^n}\), we say that \(\mathrm {b}\) is a symmetric difference product on \(\mathbb {R}^{2^n}\), or a \(\bigtriangleup \)-product on \(\mathbb {R}^{2^n}\) for short, if there exists a function \(\sigma :\mathcal {P}(n)\times \mathcal {P}(n)\rightarrow \mathbb {R}\) such that

$$\begin{aligned} \sigma (K,\emptyset )=\sigma (\emptyset ,K)=1\, \text { for all }K\in \mathcal {P}(n) \end{aligned}$$
(35)

and

$$\begin{aligned} \mathrm {b}(e_K,e_H)=e_{K\bigtriangleup H}\sigma (K,H)\, \text { for all } K,H\in \mathcal {P}(n). \end{aligned}$$
(36)

If this is the case, we say that the \(\bigtriangleup \)-product \(\mathrm {b}\) is induced by \(\sigma \), and we write \(\mathrm {b}=\mathcal {B}(\sigma )\). For simplicity, we use also the symbol \(v\cdot _\sigma w\) in place of \(\mathrm {b}(v,w)\) and, if there is no possibility of confusion, we omit ‘\(\,\cdot _\sigma \)’ writing simply vw.

Let P be a real vector space, equipped with a product \(p:P\times P\rightarrow P\) making P a real algebra. We say that the real algebra (Pp) is a symmetric difference algebra, or a \(\bigtriangleup \)-algebra for short, if it is isomorphic to some \(\mathbb {R}^{2^n}\) equipped with a \(\bigtriangleup \)-product. \(\square \)

Evidently, each \(\bigtriangleup \)-product is induced by a unique function \(\sigma \), and each function \(\sigma :\mathcal {P}(n)\times \mathcal {P}(n)\rightarrow \mathbb {R}\) satisfying (35) defines a \(\bigtriangleup \)-product on \(\mathbb {R}^{2^n}\). By Assumption 2.1, (35) and (36), we have that \(e_\emptyset =1\) is the unity of \(\mathbb {R}^{2^n}\), and \(e_K^2=\sigma (K,K)\in \mathbb {R}\) for all \(K\in \mathcal {P}(n)\); consequently, a necessary condition for a \(2^{n}\)-dimensional real algebra to be a \(\bigtriangleup \)-algebra is that it has a unity e and a vector basis \(\{v_K\}_{K\in \mathcal {P}(n)}\) such that, for each \(K\in \mathcal {P}(n)\), \(v_K^2\) belongs to the vector subspace of A generated by e.

As we will see in the next remark, the notion of \(\bigtriangleup \)-product includes several important classical products on \(\mathbb {R}^{2^n}\). Moreover, all the real algebras with unity of dimension 1 and 2 are \(\bigtriangleup \)-algebras. On the contrary, for each \(n\ge 2\), there exist \(2^n\)-dimensional real algebras with unity which are not \(\bigtriangleup \)-algebras.

Examples 2.29

(1) Each real Clifford algebra \( C \ell (p,q)\) is a \(\bigtriangleup \)-algebra, including quaternions \(\mathbb {H}= C \ell (0,2)\). When \(n=3\), another example of associative \(\bigtriangleup \)-algebra is the one of dual quaternions, see [13] for the definition. The algebra \(\mathbb {O}\) of octonions and the algebra \(\mathbb {S}\mathbb {O}\) of split-octonions are examples of non-associative \(\bigtriangleup \)-algebras, see [13].

(2) Up to isomorphism, the unique real algebra with unity of dimension 1 is \(\mathbb {R}\), which is a \(\bigtriangleup \)-algebra. All the real algebras with unity of dimension 2 are \(\bigtriangleup \)-algebras as well. Suppose that \(\mathbb {R}^2\) is equipped with a product such that 1 is its neutral element and \(e_1^2=\alpha +\beta e_1\) for some \(\alpha ,\beta \in \mathbb {R}\). Define \(v:=\beta -2e_1\). Since \(v^2=4\alpha +\beta ^2\) belongs to \(\mathbb {R}\) and \(\{1,v\}\) is a vector basis of \(\mathbb {R}^2\), it follows that \(\mathbb {R}^2\) equipped with such a product is isomorphic to \(\mathbb {R}^2\) equipped with the \(\bigtriangleup \)-product induced by the function \(\sigma :\mathcal {P}(1)\times \mathcal {P}(1)\rightarrow \mathbb {R}\) such that \(\sigma (\{1\},\{1\})=4\alpha +\beta ^2\).

Let \(n\ge 2\). Consider the product on \(\mathbb {R}^{2^n}\) such that \(e_\emptyset =1\) is its neutral element, \(e_Ke_H=0\) if \(K,H\in \mathcal {P}(n)\setminus \{\emptyset \}\) with \(K\ne H\), and \(e_K^2=-\frac{2}{2^n-1}\sum _{H\in \mathcal {P}(n)\setminus \{\emptyset \}}e_H\) for all \(K\in \mathcal {P}(n)\setminus \{\emptyset \}\). If \(v=\sum _{K\in \mathcal {P}(n)}e_Ka_K\) is a generic element of \(\mathbb {R}^{2^n}\) then

$$\begin{aligned} \textstyle v^2=a_\emptyset ^2+\sum _{K\in \mathcal {P}(n)\setminus \{\emptyset \}}e_K\big (2a_\emptyset a_K-\frac{2}{2^n-1}\sum _{H\in \mathcal {P}(n)\setminus \{\emptyset \}}a_H^2\big ). \end{aligned}$$

By simple computations, we see that \(v^2\in \mathbb {R}\) if and only if either \(v\in \mathbb {R}\) or \(v=\lambda \sum _{K\in \mathcal {P}(n)}e_K\) for some \(\lambda \in \mathbb {R}\). It follows that there exist at most two linearly independent vectors of \(\mathbb {R}^{2^n}\) whose squares are real. Consequently, \(\mathbb {R}^{2^n}\) equipped with the mentioned product is not a \(\bigtriangleup \)-algebra. \(\square \)

Other very interesting examples of \(\bigtriangleup \)-algebras can be constructed via tensor products.

Examples 2.30

Let \(n,m\in \mathbb {N}^*\). Denote \(\{e'_K\}_{K\in \mathcal {P}(n)}\) the fixed real vector basis of \(\mathbb {R}^{2^n}\), and \(\{e''_H\}_{H\in \mathcal {P}(m)}\) the fixed real vector basis of \(\mathbb {R}^{2^m}\). Recall that \(e'_\emptyset =1\in \mathbb {R}^{2^n}\) and \(e''_\emptyset =1\in \mathbb {R}^{2^m}\). Given any \(L\in \mathcal {P}(n+m)\), define \(L_m\in \mathcal {P}(m)\) and \(L_m^*\in \mathcal {P}(n)\) by setting \(L_m:=L\cap \{1,\ldots ,m\}\) and \(L_m^*:=\{l\in \mathbb {N}^*:l+m\in L\}\). Write the elements x of \(\mathbb {R}^{2^n}\otimes \mathbb {R}^{2^m}\) as follows:

$$\begin{aligned} \textstyle x=\sum _{H\in \mathcal {P}(m)}e''_H(\sum _{K\in \mathcal {P}(n)}e'_Kr_{H,K})=\sum _{H\in \mathcal {P}(m),K\in \mathcal {P}(n)}e''_He'_Kr_{H,K} \end{aligned}$$

for \(r_{H,K}\in \mathbb {R}\), where \(e''_He'_K:=e'_K\otimes e''_H\). Identify \(\mathbb {R}^{2^{n+m}}\) with \(\mathbb {R}^{2^n}\otimes \mathbb {R}^{2^m}\) and define the real vector basis \(\{e_L\}_{L\in \mathcal {P}(n+m)}\) of \(\mathbb {R}^{2^{n+m}}\) by \(e_L:=e''_{L_m}e'_{L_m^*}\). In this way, we can write \(x=\sum _{L\in \mathcal {P}(n+m)}e_Lr_L\), where \(r_L:=r_{L_m,L_m^*}\).

Choose a \(\bigtriangleup \)-product \(\mathrm {b}=\mathcal {B}(\sigma )\) on \(\mathbb {R}^{2^n}\) and a \(\bigtriangleup \)-product \(\mathrm {c}=\mathcal {B}(\tau )\) on \(\mathbb {R}^{2^m}\). Define the function \(\sigma \otimes \tau :\mathcal {P}(n+m)\times \mathcal {P}(n+m)\rightarrow \mathbb {R}\) and the \(\bigtriangleup \)-product \(\mathrm {b}\otimes \mathrm {c}\) on \(\mathbb {R}^{2^{n+m}}\) as follows:

$$\begin{aligned} (\sigma \otimes \tau )(L,M)&:=\sigma (L_m^*,M_m^*)\tau (L_m,M_m), \end{aligned}$$
(37)
$$\begin{aligned} \mathrm {b}\otimes \mathrm {c}&:=\mathcal {B}(\sigma \otimes \tau ). \end{aligned}$$
(38)

We call \((\mathbb {R}^{2^{n+m}},\mathrm {b}\otimes \mathrm {c})\) tensor product of the \(\bigtriangleup \)-algebras \((\mathbb {R}^{2^n},\mathrm {b})\) and \((\mathbb {R}^{2^m},\mathrm {c})\). Note that, given \(L,M\in \mathcal {P}(n+m)\), it holds:

$$\begin{aligned}&(e'_{L_m^*}\otimes e''_{L_m})\cdot _{\sigma \otimes \tau }(e'_{M_m^*}\otimes e''_{M_m})\\&\qquad =e_L\cdot _{\sigma \otimes \tau }e_M=e_{L\bigtriangleup M}\sigma (L_m^*,M_m^*)\tau (L_m,M_m)\\&\qquad =(e'_{(L\bigtriangleup M)^*_m}\otimes e''_{(L\bigtriangleup M)_m})\sigma (L_m^*,M_m^*)\tau (L_m,M_m)\\&\qquad =(e'_{L_m^*\bigtriangleup M_m^*}\otimes e''_{L_m\bigtriangleup M_m})\sigma (L_m^*,M_m^*)\tau (L_m,M_m)\\&\qquad =(e'_{L_m^*\bigtriangleup M_m^*}\sigma (L_m^*,M_m^*))\otimes (e''_{L_m\bigtriangleup M_m}\tau (L_m,M_m))\\&\qquad =(e'_{L^*_m}\cdot _{\sigma }e'_{M^*_m})\otimes (e''_{L_m}\cdot _{\tau }e''_{M_m}).\, \end{aligned}$$

\(\square \)

Note that the real algebra \(\mathbb {C}\) of complex numbers coincides with \(\mathbb {R}^2\) equipped with the \(\bigtriangleup \)-product \(\eta :\mathcal {P}(1)\times \mathcal {P}(1)\rightarrow \mathbb {R}\) such that \(\eta (\{1\},\{1\}):=-1\).

Definition 2.31

Given any \(n\in \mathbb {N}^*\), we denote \(\sigma _\otimes ^n:\mathcal {P}(n)\times \mathcal {P}(n)\rightarrow \mathbb {R}\) the n-times iterated tensor product of \(\eta :\mathcal {P}(1)\times \mathcal {P}(1)\rightarrow \mathbb {R}\) with itself, i.e. \(\sigma _\otimes ^1:=\eta \) and \(\sigma _\otimes ^n:=\sigma _\otimes ^{n-1}\otimes \eta \) if \(n\ge 2\). We say that \(\mathrm {b}_\otimes ^n:=\mathcal {B}(\sigma _\otimes ^n)\) is the tensor product on \(\mathbb {C}^{\otimes n}=\mathbb {R}^{2^n}\), and \(\mathbb {C}^{\otimes n}\) equipped with \(\mathrm {b}_\otimes ^n\) is the \(n^{\mathrm {th}}\)-tensor power of \(\mathbb {C}\). \(\square \)

Lemma 2.32

For all \(n\in \mathbb {N}^*\) and for all \(K,H\in \mathcal {P}(n)\), it holds \(\sigma _\otimes ^n(K,H)=(-1)^{|K\cap H|}\). In particular, the \(n^{\mathrm {th}}\)-tensor power \(\mathbb {C}^{\otimes n}\) of \(\mathbb {C}\) is commutative and associative.

Proof

Let us prove this assertion by induction on \(n\in \mathbb {N}^*\). The case \(n=1\) is evident, because \(\sigma _\otimes ^1=\eta \) and \(\eta \) has the required property. Let \(n\ge 2\). By induction, there exists a real vector basis \(\{e_{H'}\}_{H'\in \mathcal {P}(n-1)}\) of \(\mathbb {R}^{2^{n-1}}\) such that \(\sigma _\otimes ^{n-1}(K',H')=(-1)^{|K'\cap H'|}\) for all \(K',H'\in \mathcal {P}(n-1)\). By (37), we have that \(\sigma _\otimes ^n(K,H)=\sigma _\otimes ^{n-1}(K_1^*,H_1^*)\eta (K_1,L_1)=(-1)^{|K_1^*\cap H_1^*|+|K_1\cap L_1|}\). Since \(K_1^*\cap H_1^*=(K\cap H)^*_1\) and \(K_1\cap H_1=(K\cap H)_1\), we easily deduce that \(|K_1^*\cap H_1^*|+|K_1\cap L_1|=|K\cap H|\), as desired. \(\square \)

Assumption 2.33

Throughout the remaining part of this section, we equip \(\mathbb {R}^{2^n}\) with a \(\bigtriangleup \)-product \(\mathrm {b}=\mathcal {B}(\sigma )\), and the tensor product \(A\otimes \mathbb {R}^{2^n}\) with the following product extending \(\mathrm {b}\):

$$\begin{aligned} \textstyle \big (\sum _{H\in \mathcal {P}(n)}e_Ha_H\big )\cdot _\sigma \big (\sum _{L\in \mathcal {P}(n)}e_Lb_L\big ):=\sum _{H,L\in \mathcal {P}(n)}(e_H\cdot _\sigma e_L)(a_Hb_L), \end{aligned}$$
(39)

where \(a_Hb_L\) is the product of \(a_H\) and \(b_L\) in A. For simplicity, for each \(\xi ,\eta \in A\otimes \mathbb {R}^{2^n}\), we also write \(\xi \eta \) in place of \(\xi \cdot _\sigma \eta \).

Note that if, for each \(K\in \mathcal {P}(n)\), \(\mathscr {D}(K)\) denotes the set

$$\begin{aligned} \mathscr {D}(K):=\big \{(K_1,K_2,K_3) \in \mathcal {P}(n)^3 \,:\, K_1 \cap K_2=\emptyset , K_1 \cup K_2=K, K \cap K_3=\emptyset \big \}, \end{aligned}$$

then \(\big (\sum _{H\in \mathcal {P}(n)}e_Ha_H\big )\big (\sum _{L\in \mathcal {P}(n)}e_Lb_L\big )=\sum _{K\in \mathcal {P}(n)}e_Kc_K\), where

$$\begin{aligned} \textstyle c_K=\sum _{(K_1,K_2,K_3) \in \mathscr {D}(K)}a_{K_1 \cup K_3}b_{K_2 \cup K_3}\sigma (K_1 \cup K_3,K_2 \cup K_3). \end{aligned}$$
(40)

Indeed, we have:

$$\begin{aligned}&\textstyle \sum _{H,L \in \mathcal {P}(n)}(e_He_L)(a_Hb_L)=\sum _{H,L \in \mathcal {P}(n)}e_{H \bigtriangleup L}a_Hb_L\sigma (H,L)\\&\textstyle =\sum _{K \in \mathcal {P}(n)}e_K\sum _{(K_1,K_2,K_3) \in \mathscr {D}(K)}a_{K_1 \cup K_3}b_{K_2 \cup K_3}\sigma (K_1 \cup K_3,K_2 \cup K_3). \end{aligned}$$

In general the pointwise product of two slice functions is not a slice function. For instance, if \(n=1\) and \(f,g:\mathbb {H}\rightarrow \mathbb {H}\) are the slice functions defined by \(f\equiv i\) and \(g(x)=x\), then \((fg)(x)=ix\) is not slice. Otherwise, by Proposition 2.9 and Corollary 2.13, it would follow that \((fg)(x)=xi\) for all \(x\in \mathbb {H}\), which is impossible being \((fg)(j)=ji\ne ij\). On the contrary, the pointwise product of two stem functions is still a stem function.

Lemma 2.34

Let \(F,G:D\rightarrow A \otimes \mathbb {R}^{2^n}\) be stem functions and let \(F\cdot _\sigma G:D \rightarrow A \otimes \mathbb {R}^{2^n}\) be the pointwise product of F and G w.r.t. \(\mathrm {b}=\mathcal {B}(\sigma )\), that is \((F\cdot _\sigma G)(z):=F(z)\cdot _\sigma G(z)\) for all \(z\in D\). Then \(F\cdot _\sigma G\) is still a stem function.

Proof

Write FG in place of \(F\cdot _\sigma G\), for short. By (40), if \(F=\sum _{H \in \mathcal {P}(n)}e_HF_H\), \(G=\sum _{L \in \mathcal {P}(n)}e_LG_L\) and \(FG=\sum _{K\in \mathcal {P}(n)}e_K(FG)_K\), then

$$\begin{aligned} \textstyle (FG)_K=\sum _{(K_1,K_2,K_3) \in \mathscr {D}(K)}F_{K_1 \cup K_3}G_{K_2 \cup K_3}\sigma (K_1 \cup K_3,K_2 \cup K_3). \end{aligned}$$

Choose \(K \in \mathcal {P}(n)\), \(h \in \{1,\ldots ,n\}\) and \(z \in D\). Note that, for each \((K_1,K_2,K_3) \in \mathscr {D}(K)\), the integers \(|(K_1 \cup K_3) \cap \{h\}|+|(K_2 \cup K_3) \cap \{h\}|\) and \(|K \cap \{h\}|\) have the same parity. Consequently, we have

$$\begin{aligned} (FG)_K({\overline{z}}^h)&=\textstyle \sum _{(K_1,K_2,K_3) \in \mathscr {D}(K)}F_{K_1 \cup K_3}({\overline{z}}^h) \, G_{K_2 \cup K_3}({\overline{z}}^h) \, \sigma (K_1 \cup K_3,K_2 \cup K_3)\\&=\textstyle (-1)^{|K \cap \{h\}|}\sum _{(K_1,K_2,K_3) \in \mathscr {D}(K)}F_{K_1 \cup K_3}(z) \, G_{K_2 \cup K_3}(z)\times \\&\quad \times \sigma (K_1 \cup K_3,K_2 \cup K_3)=\textstyle (-1)^{|K \cap \{h\}|}(FG)_K(z), \end{aligned}$$

so FG is a stem function, as desired. \(\square \)

Thanks to the latter lemma, we can define a product on the class of slice functions.

Definition 2.35

Let \(f,g:\Omega _D \rightarrow A\) be slice functions with \(f=\mathcal {I}(F)\) and \(g=\mathcal {I}(G)\). We define the slice product \(f\cdot _\sigma g:\Omega _D\rightarrow A\) of f and g by \(f\cdot _\sigma g:=\mathcal {I}(F\cdot _\sigma G)\). Moreover, we say that the slice product \(f\cdot g=f\cdot _\sigma g\) is induced by \(\mathrm {b}\), or by \(\sigma \). If there is no possibility of confusion, we simply write FG and \(f\cdot g\) in place of \(F\cdot _\sigma G\) and \(f\cdot _\sigma g\), respectively. \(\square \)

We specialize the preceding definition as follows.

Definition 2.36

We call slice tensor product on \(\mathcal {S}(\Omega _D,A)\) the product on \(\mathcal {S}(\Omega _D,A)\) induced by the tensor product \(\mathrm {b}_\otimes ^n=\mathcal {B}(\sigma _\otimes ^n)\). Given \(f,g\in \mathcal {S}(\Omega _D,A)\), we say that \(f\cdot _{\sigma _\otimes ^n} g\) is the slice tensor product of f and g. \(\square \)

Assumption 2.37

In what follows, we use the symbol ‘\(\,\odot \)’ to denote ‘\(\,\cdot _{\sigma _\otimes ^n}\,\)’.

Corollary 2.15 imply at once the following fact.

Corollary 2.38

The pairs \((\mathrm {Stem}(D,A\otimes \mathbb {R}^{2^n}),\cdot _\sigma )\) and \((\mathcal {S}(\Omega _D,A),\cdot _\sigma )\) are real algebras, and \(\mathcal {I}\) is a real algebra isomorphism between them.

Let us introduce the concepts of slice polynomial functions associated with \(\mathrm {b}=\mathcal {B}(\sigma )\), and of hypercomplex \(\bigtriangleup \)-product.

Definition 2.39

Let \(\mathrm {b}=\mathcal {B}(\sigma )\) be a \(\triangle \)-product on \(\mathbb {R}^{2^n}\). Given any \(k\in \{1,\ldots ,n\}\) and \(m\in \mathbb {N}\), we define the slice function \(x_k^{{\scriptscriptstyle \bullet } m}:(Q_A)^n\rightarrow A\) as the function constantly equal to 1 if \(m=0\), as the \(k^{\text {th}}\)-coordinate function \(x_k:(Q_A)^n\rightarrow A\) if \(m=1\) and as the m-times iterated slice product of \(x_k:(Q_A)^n\rightarrow A\) with itself w.r.t. \(\mathrm {b}\) if \(m\ge 2\), i.e. \(x_k^{{\scriptscriptstyle \bullet } 0}:\equiv 1\) and \(x_k^{{\scriptscriptstyle \bullet } m}:=x_k^{{\scriptscriptstyle \bullet } m-1}\cdot _\sigma x_k\) if \(m\ge 1\). We say that a function \(P:(Q_A)^n\rightarrow A\) is slice monomial w.r.t. \(\mathrm {b}\), or w.r.t. \(\sigma \), if there exist \(\ell =(\ell _1,\ldots ,\ell _n)\in \mathbb {N}\) and \(a\in A\) such that

$$\begin{aligned} P=x_1^{{\scriptscriptstyle \bullet }\ell _1}\cdot _\sigma (x_2^{{\scriptscriptstyle \bullet }\ell _2}\cdots (x_{m-1}^{{\scriptscriptstyle \bullet }\ell _{m-1}}\cdot _{\sigma } (x_m^{{\scriptscriptstyle \bullet }\ell _m} \cdot _{\sigma } a)) \ldots ), \end{aligned}$$

where \(a\in A\) is identified with the slice function from \((Q_A)^n\) to A constantly equal to a. If P has this form and there is no possibility of confusion, then we denote P as \(x^{{\scriptscriptstyle \bullet }\ell }\cdot a\). We call \(P:(Q_A)^n\rightarrow A\) a slice polynomial function w.r.t. \(\mathrm {b}\), or w.r.t. \(\sigma \) if it is a finite sum of slice monomial functions w.r.t. \(\mathrm {b}\), or w.r.t. \(\sigma \).

The restriction of a slice monomial (respectively polynomial) function w.r.t. \(\mathrm {b}\), or w.r.t. \(\sigma \), to \(\Omega _D\) is said to be a slice monomial (respectively polynomial) on \(\Omega _D\) w.r.t. \(\mathrm {b}\), or w.r.t. \(\sigma \). \(\square \)

Definition 2.40

We say that the \(\bigtriangleup \)-product \(\mathrm {b}=\mathcal {B}(\sigma )\) on \(\mathbb {R}^{2^n}\) is hypercomplex if it satisfies the following two conditions:

$$\begin{aligned} e_k^2=-1 \end{aligned}$$
(41)

for all \(k\in \{1,\ldots ,n\}\), and

$$\begin{aligned} e_K=e_{k_1}(e_{k_2}\cdots (e_{k_{s-1}}e_{k_s})\ldots ) \end{aligned}$$
(42)

for all \(K\in \mathcal {P}(n)\setminus \{\emptyset \}\) with \(K=\{k_1,\ldots ,k_s\}\) and \(k_1<\cdots <k_s\), which are equivalent to \(\sigma (\{k\},\{k\})=-1\) and \(\sigma (\{k_1\},\{k_2,\ldots ,k_s\})\cdots \sigma (\{k_{s-2}\},\{k_{s-1},k_s\})\sigma (\{k_{s-1}\},\{k_s\})=1\), respectively. We say that a real algebra is a hypercomplex \(\bigtriangleup \)-algebra if it is isomorphic to some \(\mathbb {R}^{2^n}\) equipped with a hypercomplex \(\bigtriangleup \)-product. \(\square \)

Examples 2.41

The real algebras \(\mathbb {C}^{\otimes n}\), \(\mathbb {R}_q= C \ell (0,q)\) and all their finite tensor products are hypercomplex \(\bigtriangleup \)-algebras. The real algebra \(\mathbb {C}=\mathbb {C}^{\otimes 1}=\mathbb {R}_1\) of complex numbers is the unique hypercomplex \(\bigtriangleup \)-algebra of dimension 2. The Clifford algebra \( C \ell (1,0)\) is an example of \(\bigtriangleup \)-algebra, which is not hypercomplex; indeed it has no imaginary units. A natural question is to understand whether the \(\bigtriangleup \)-algebra \( C \ell (p,q)\) is hypercomplex when \(p\ge 1\). This problem seems to be not so easy to settle. We recall several relations existing among Clifford algebras; for instance, \( C \ell (4,q-4)\) and \(\mathbb {R}_q\) are isomorphic if \(q\ge 4\), see [19,  §16.4]. \(\square \)

Lemma 2.42

The unique hypercomplex, commutative and associative \(\bigtriangleup \)-product on \(\mathbb {R}^{2^n}\) is the tensor product \(\mathrm {b}_\otimes ^n\).

Proof

By Lemma 2.32, we know that \(\mathrm {b}_\otimes ^n\) is hypercomplex, commutative and associative. Suppose \(\mathrm {b}=\mathcal {B}(\sigma )\) is a \(\bigtriangleup \)-product on \(\mathbb {R}^{2^n}\) which is hypercomplex, commutative and associative. By (42), the commutativity and the associativity, we have that \(e_K\cdot _\sigma e_H=e_{K\bigtriangleup H}\cdot _\sigma e_{\ell _1}^2\cdot _\sigma \cdots \cdot _\sigma e_{\ell _q}^2\), where \(\ell _1,\ldots ,\ell _q\) are the elements of \(K\cap H\) if \(K\cap H\ne \emptyset \), and ‘\(e_{\ell _1}^2\cdot _\sigma \cdots \cdot _\sigma e_{\ell _q}^2\)’ is omitted if \(K\cap H=\emptyset \). Using (41) and Lemma 2.32 again, we deduce that \(e_K\cdot _\sigma e_H=e_{K\bigtriangleup H}(-1)^{|K\cap H|}\) and \(\mathrm {b}=\mathrm {b}_\otimes ^n\). \(\square \)

The next result describes the ‘algebraic relevance’ of hypercomplex \(\bigtriangleup \)-algebras in the context of slice functions. It asserts that, if the \(\bigtriangleup \)-product \(\mathrm {b}=\mathcal {B}(\sigma )\) is hypercomplex, then the notions of polynomial function, pointwise defined in Definition 2.7, and of slice polynomial function defined in Definition 2.39 coincide.

Lemma 2.43

If the \(\bigtriangleup \)-product \(\mathrm {b}\) on \(\mathbb {R}^{2^n}\) is hypercomplex (for instance, \(\mathrm {b}=\mathrm {b}_\otimes ^n\)), then a function \(f:\Omega _D\rightarrow A\) is polynomial if and only if it is slice polynomial w.r.t. \(\mathrm {b}\). More precisely, if \(f(x)=\sum _{\ell \in L}x^\ell a_\ell \) for some finite subset L of \(\mathbb {N}^n\) and \(a_\ell \in A\), then \(f=\sum _{\ell \in L}x^{{\scriptscriptstyle \bullet }\ell }\cdot a_\ell \) on \(\Omega _D\).

Proof

Let \(\ell =(\ell _1,\ldots ,\ell _n)\in \mathbb {N}^n\) and let \(F^{(\ell )}:D\rightarrow A\otimes \mathbb {R}^{2^n}\) be the stem function inducing \(x^{{\scriptscriptstyle \bullet }\ell }\cdot a_\ell \). Given \(h\in \{1,\ldots ,n\}\), denote \(G^{(h)}:D\rightarrow A\otimes \mathbb {R}^{2^n}\) the stem function inducing \(x_h^{\bullet \ell _h}:\Omega _D\rightarrow A\). Let \(z=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\in D\), and let \(p_{\ell _h}\) and \(q_{\ell _h}\) be the real polynomials defined in Definition 2.8. Since \(\mathrm {b}\) satisfies (41) and (42), we have that \(G^{(h)}(z)=p_{\ell _h}(\alpha _h,\beta _h)+e_hq_{\ell _h}(\alpha _h,\beta _h)\) and

$$\begin{aligned} F^{(\ell )}(z)&\textstyle =[(G^{(h)}(z))_{h=1}^n,a_\ell ]=[(p_{\ell _h}(\alpha _h,\beta _h)+e_hq_{\ell _h}(\alpha _h,\beta _h))_{h=1}^n,a_\ell ]\\&\textstyle =\sum _{K\in \mathcal {P}(n)}\big (\prod _{h\in \{1,\ldots ,n\}\setminus K}p_{\ell _h}(\alpha _h,\beta _h)\big )[(e_hq_{\ell _h}(\alpha _h,\beta _h))_{h\in K},a_\ell ]\\&\textstyle =\sum _{K\in \mathcal {P}(n)}e_K\big (\big (\prod _{h\in \{1,\ldots ,n\}\setminus K}p_{\ell _h}(\alpha _h,\beta _h)\big )\big (\prod _{h\in K}q_{\ell _h}(\alpha _h,\beta _h)\big )a_\ell \big ). \end{aligned}$$

By Proposition 2.9, it follows that \(x^{{\scriptscriptstyle \bullet }\ell }\cdot a_\ell =\mathcal {I}(F^{(\ell )})=x^\ell a_\ell \). Consequently, \(f=\sum _{\ell \in L}x^{{\scriptscriptstyle \bullet }\ell }\cdot a_\ell \). \(\square \)

\(\mathbb {C}_J\)-slice preserving, slice preserving and circular functions

Definition 2.44

Given any function \(f:\Omega _D \rightarrow A\) and \(J \in {\mathbb {S}}_A\), we denote \(\Omega _D(J)\) the intersection \(\Omega _D \cap (\mathbb {C}_J)^n\), and \(f_J:\Omega _D(J)\rightarrow A\) the restriction of f to \(\Omega _D(J)\). \(\square \)

Definition 2.45

Let \(f:\Omega _D\rightarrow A\) be a function. Given \(J\in {\mathbb {S}}_A\), we say that f is a \(\mathbb {C}_J\)-slice preserving function if f is a slice function and \(f(\Omega _D(J))\subset \mathbb {C}_J\). We denote \(\mathcal {S}_{\mathbb {C}_J}(\Omega _D,A)\) the subset of \(\mathcal {S}(\Omega _D,A)\) of all \(\mathbb {C}_J\)-slice preserving functions from \(\Omega _D\) to A.

We say that f is slice preserving if it is a slice function and the stem function \(F=\sum _{K\in \mathcal {P}(n)}e_KF_K\) inducing f has the following property: each component \(F_K\) of F is real-valued, that is \(F_K(D)\subset \mathbb {R}\) for all \(K\in \mathcal {P}(n)\). We denote \(\mathcal {S}_\mathbb {R}(\Omega _D,A)\) the subset of \(\mathcal {S}(\Omega _D,A)\) of all slice preserving functions from \(\Omega _D\) to A. \(\square \)

Lemma 2.46

Let \(F:\Omega _D\rightarrow A\otimes \mathbb {R}^{2^n}\) be a stem function with \(F=\sum _{K\in \mathcal {P}(n)}e_KF_K\) and let \(f:=\mathcal {I}(F):\Omega _D\rightarrow A\) be the corresponding slice function. The following assertions hold.

  • \((\mathrm {i})\) Given \(J \in {\mathbb {S}}_A\), f belongs to \(\mathcal {S}_{\mathbb {C}_J}(\Omega _D,A)\) if and only if \(F_K(D)\subset \mathbb {C}_J\) for all \(K\in \mathcal {P}(n)\).

  • \((\mathrm {ii})\) Suppose that there exist \(I,J\in {\mathbb {S}}_A\) such that \(I\ne \pm J\). Then f is slice preserving if and only if it is \(\mathbb {C}_K\)-slice preserving for \(K\in \{I,J\}\), or equivalently \(\mathcal {S}_\mathbb {R}(\Omega _D,A)=\bigcap _{K\in {\mathbb {S}}_A}\mathcal {S}_{\mathbb {C}_K}(\Omega _D,A)\).

Proof

Since \(\mathbb {C}_J\) is a real subalgebra of A, if \(F_K(D)\subset \mathbb {C}_J\) for all \(K\in \mathcal {P}(n)\), then Definition 2.5 implies at once that \(f(\Omega _D(J))\subset \mathbb {C}_J\). Suppose now \(f(\Omega _D(J))\subset \mathbb {C}_J\) and apply formula (8) to f with \(I_1=\ldots =I_n=J\). We obtain immediately that \(F_K(D)\subset \mathbb {C}_J\) for all \(K\in \mathcal {P}(n)\). This proves \((\mathrm {i})\). Let us show \((\mathrm {ii})\). Recall that \(\mathbb {R}\subset \mathbb {C}_J\) for all \(J\in {\mathbb {S}}_A\). As a consequence, the preceding point \((\mathrm {i})\) implies that \(\mathcal {S}_\mathbb {R}(\Omega _D,A)\subset \bigcap _{K\in {\mathbb {S}}_A}\mathcal {S}_{\mathbb {C}_K}(\Omega _D,A)\). Finally, if \(f\in \mathcal {S}_{\mathbb {C}_I}(\Omega _D,A)\cap \mathcal {S}_{\mathbb {C}_J}(\Omega _D,A)\), then using again above point \((\mathrm {i})\) we deduce that \(F_K(D)\subset \mathbb {C}_I\cap \mathbb {C}_J\). Thanks to the Independence Lemma [3,  p. 224], we know that \(\mathbb {C}_I\cap \mathbb {C}_J=\mathbb {R}\), and we are done. \(\square \)

Since each plane \(\mathbb {C}_J\) is a real subalgebra of A, it follows at once:

Lemma 2.47

Let \(\mathrm {b}=\mathcal {B}(\sigma )\) be any \(\bigtriangleup \)-product on \(\mathbb {R}^{2^n}\) and let \(J\in {\mathbb {S}}_A\). The sets \(\mathcal {S}_\mathbb {R}(\Omega _D,A)\) and \(\mathcal {S}_{\mathbb {C}_J}(\Omega _D,A)\) are real subalgebras of \((\mathcal {S}(\Omega _D,A),\cdot _\sigma )\).

The next result concerns the relation between slice tensor and pointwise products. Given two functions \(f,g:\Omega _D\rightarrow A\), we indicate \(fg:\Omega _D\rightarrow A\) the pointwise product of f and g, i.e. \((fg)(x):=f(x)g(x)\) for all \(x\in \Omega _D\), where f(x)g(x) is the product of f(x) and g(x) in A.

Proposition 2.48

Let \(f,g\in \mathcal {S}_{\mathbb {C}_J}(\Omega _D,A)\) for some \(J\in {\mathbb {S}}_A\), and let \(a\in A\). Identify a with the function from \(\Omega _D\) (or from \(\Omega _D(J)\)) to A constantly equal to a. The following holds:

  • \((\mathrm {i})\) \((f\odot (g\odot a))(x)=f(x)g(x)a\) for all \(x\in \Omega _D(J)\). Equivalently, \((f\odot (g\odot a))_J=f_Jg_Ja\).

  • \((\mathrm {ii})\) \(f\odot (g\odot a)=(f\odot g)\odot a\,\) on the whole \(\Omega _D\).

Proof

First note that \((\mathrm {i})\) implies \((\mathrm {ii})\). Indeed, applying \((\mathrm {i})\) twice (with \(a=1\) and \(g\equiv 1\)), Lemma 2.47 and Artin’s theorem, one obtains \((f\odot g)_J=f_Jg_J\) and \(((f\odot g)\odot a)_J=(f\odot g)_Ja=f_Jg_Ja=(f\odot (g\odot a))_J\). Corollary 2.13 implies \((\mathrm {ii})\). Let us prove \((\mathrm {i})\). Let \(F=\sum _{K\in \mathcal {P}(n)}e_KF_K\) and \(G=\sum _{H\in \mathcal {P}(n)}e_HG_H\) be the stem functions inducing f and g, respectively. Let \(C=\sum _{L\in \mathcal {P}(n)}e_LC_L:D\rightarrow A\otimes \mathbb {R}^{2^n}\) be the stem function constantly equal to a, i.e. \(C_\emptyset =a\) on D and \(C_L=0\) on D for all \(L\in \mathcal {P}(n)\setminus \{\emptyset \}\). Evidently, \(\mathcal {I}(C)=a\). Consider \(z=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\in D\) and \(x=(\alpha _1+J\beta _1,\ldots ,\alpha _n+J\beta _n)\in \Omega _D(J)\). Let \(\mathrm {J}:=(J,\ldots ,J)\in ({\mathbb {S}}_A)^n\). By Lemma 2.32, we have that

$$\begin{aligned} \textstyle F\odot (G\odot C)=\sum _{K,H\in \mathcal {P}(n)}e_{K \bigtriangleup H}(-1)^{|K\cap H|}F_K(G_Ha). \end{aligned}$$

Lemma 2.46\((\mathrm {i})\) implies that \(F_K(z),G_H(z)\in \mathbb {C}_J\), so the elements \(F_K(z)\), \(G_H(z)\) and J of A commute and associate. By Artin’s theorem, it follows that

$$\begin{aligned} (f\odot (g\odot a))(x)&\textstyle =\sum _{K,H \in \mathcal {P}(n)}[\mathrm {J}_{K \bigtriangleup H},(-1)^{|K\cap H|}F_K(z)G_H(z)a]\\&\textstyle =\sum _{K,H \in \mathcal {P}(n)}J^{|K \bigtriangleup H|}(-1)^{|K\cap H|}F_K(z)G_H(z)a\\&\textstyle =\sum _{K,H \in \mathcal {P}(n)}J^{|K \bigtriangleup H|}J^{2|K\cap H|}F_K(z)G_H(z)a\\&\textstyle =\sum _{K,H \in \mathcal {P}(n)}J^{|K \bigtriangleup H|+2|K\cap H|}F_K(z)G_H(z)a\\&\textstyle =\sum _{K,H \in \mathcal {P}(n)}J^{|K|+|H|}F_K(z)G_H(z)a\\&\textstyle =(\sum _{K\in \mathcal {P}(n)}J^{|K|}F_K(z))(\sum _{H\in \mathcal {P}(n)}J^{|H|}G_H(z))a=f(x)g(x)a. \end{aligned}$$

The proof is complete. \(\square \)

Lemma 2.49

Let \(\mathrm {b}=\mathcal {B}(\sigma )\) be a commutative and associative \(\triangle \)-product on \(\mathbb {R}^{2^n}\) (for instance \(\mathrm {b}=\mathrm {b}^n_\otimes \)). Then the set \(\mathcal {S}_\mathbb {R}(\Omega _D,A)\) is contained in the center of \((\mathcal {S}(\Omega _D,A),\cdot _\sigma )\).

Proof

Let \(F=\sum _{K\in \mathcal {P}(n)}e_KF_K\), \(G=\sum _{L\in \mathcal {P}(n)}e_LG_L\) and \(H=\sum _{M\in \mathcal {P}(n)}e_MH_M\) be stem functions on D such that each \(F_K\) is real-valued. Bearing in mind the latter condition and the fact that ‘\(\,\cdot _\sigma \,\)’ is commutative and associative, we obtain:

$$\begin{aligned} FG&=\textstyle \sum _{K,L\in \mathcal {P}(n)}e_K\cdot _\sigma e_LF_KG_L=\sum _{L,K\in \mathcal {P}(n)}e_L\cdot _\sigma e_KG_LF_K=GF,\\ (FG)H&=\textstyle \sum _{K,L,M\in \mathcal {P}(n)}e_K\cdot _\sigma e_L\cdot _\sigma e_M(F_KG_L)H_M\\&=\textstyle \sum _{K,L,M\in \mathcal {P}(n)}e_K\cdot _\sigma e_L\cdot _\sigma e_MF_K(G_LH_M)=F(GH). \end{aligned}$$

Similar considerations prove also that \((GF)H=G(FH)\) and \((GH)F=G(HF)\). \(\square \)

In [9,  Remark 7] we proved that, if \(n=1\), \(f\in \mathcal {S}_\mathbb {R}(\Omega _D,A)\) and \(g\in \mathcal {S}(\Omega _D,A)\), then \(f\odot g=fg\) on the whole \(\Omega _D\). Remark 2.14 shows that in general the latter equality is false for \(n\ge 2\); indeed, the coordinate functions \(x_2\) and \(x_1\) belong to \(\mathcal {S}_\mathbb {R}(\mathbb {H}^n,\mathbb {H})\), but \(x_2x_1\) is not slice on \(\mathbb {H}^n\).

Our next results give some generalizations to several variables of the mentioned result contained in [9,  Remark 7]. First, we need a definition.

Definition 2.50

Let \(F=\sum _{K\in \mathcal {P}(n)}e_KF_K:D\rightarrow A\otimes \mathbb {R}^{2^n}\) be a stem function, let \(f=\mathcal {I}(F):\Omega _D\rightarrow A\) be the slice function induced by F and let \(H\in \mathcal {P}(n)\). We say that F is H-reduced if \(F_K=0\) on D for all \(K\in \mathcal {P}(n)\) with \(K\not \subset H\). If F is H-reduced then we say also that f is H-reduced. If \(H=\{h\}\) for some \(h\in \{1,\ldots ,n\}\), then we use the term h-reduced meaning \(\{h\}\)-reduced. We say that f is circular if it is \(\emptyset \)-reduced, namely if F is A-valued. Denote \(\mathcal {S}_c(\Omega _D,A)\) the subset of \(\mathcal {S}(\Omega _D,A)\) formed by all circular functions. \(\square \)

Lemma 2.51

Let \(\mathrm {b}=\mathcal {B}(\sigma )\) be any \(\bigtriangleup \)-product on \(\mathbb {R}^{2^n}\). Then, it holds:

  • \((\mathrm {i})\) The set \(\mathcal {S}_c(\Omega _D,A)\) is a real subalgebra of \((\mathcal {S}(\Omega _D,A),\cdot _\sigma )\). Furthermore, if \(f,g\in \mathcal {S}_c(\Omega _D,A)\), then \(f\cdot _\sigma g=fg\) on \(\Omega _D\).

  • \((\mathrm {ii})\) If \(f\in \mathcal {S}_\mathbb {R}(\Omega _D,A)\cap \mathcal {S}_c(\Omega _D,A)\) and \(g\in \mathcal {S}(\Omega _D,A)\), then \(f\cdot _\sigma g=fg\) on \(\Omega _D\).

  • \((\mathrm {iii})\) If A is associative, \(f\in \mathcal {S}(\Omega _D,A)\) and \(g\in {\mathcal {S}}_c(\Omega _D,A)\), then \(f\cdot _\sigma g=fg\) on \(\Omega _D\).

Proof

Let F and G be the stem functions inducing f and g, respectively. Point \((\mathrm {i})\) follows immediately from the fact that, for all \(z\in D\), \((F\cdot _\sigma G)(z)\) is equal to the product of \(F_\emptyset (z)\) and \(G_\emptyset (z)\) in A. Let us prove \((\mathrm {ii})\). In this case \(F=F_\emptyset \) is real-valued, \(G=\sum _{K\in \mathcal {P}(n)}e_KG_K\) is generic and

$$\begin{aligned} (f\cdot _\sigma g)(x)&\textstyle =\sum _{K\in \mathcal {P}(n)}[J_K,F_\emptyset (z)G_K(z)]=F_\emptyset (z)\sum _{K\in \mathcal {P}(n)}[J_K,G_K(z)]=f(x)g(x), \end{aligned}$$

as desired. If A is associative and \(G=G_\emptyset \), we can write

$$\begin{aligned} (f\cdot _\sigma g)(x)\textstyle =\sum _{K\in \mathcal {P}(n)}[J_K,F_K(z)G_\emptyset (z)]=\sum _{K\in \mathcal {P}(n)}[J_K,F_K(z)]G_\emptyset (z)=f(x)g(x), \end{aligned}$$

and \((\mathrm {iii})\) is proved. \(\square \)

Proposition 2.52

Let \(\mathrm {b}=\mathcal {B}(\sigma )\) be an associative and hypercomplex \(\triangle \)-product on \(\mathbb {R}^{2^n}\) (for instance \(\mathrm {b}=\mathrm {b}^n_\otimes \)), let \(f\in \mathcal {S}_\mathbb {R}(\Omega _D,A)\) and let \(g\in \mathcal {S}(\Omega _D,A)\). Suppose that there exist \(\ell \in \{1,\ldots ,n\}\) and \(H\in \mathcal {P}(n)\) such that f is \(\ell \)-reduced, g is H-reduced and \(\ell \le h\) for all \(h\in H\). Then \(f\cdot _\sigma g=fg\) on \(\Omega _D\).

Proof

Denote \(F,G\in \mathrm {Stem}(D,A\otimes \mathbb {R}^{2^n})\) the stem functions inducing f and g, respectively. Write \(F=F_\emptyset +e_\ell F_\ell \) and \(G=\sum _{K\in \mathcal {P}(n),K\subset H}e_KG_K\). Here \(F_\ell \) denotes \(F_{\{\ell \}}\). Let \(z=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\in D\) and let \(x=(\alpha _1+J_1\beta _1,\ldots ,\alpha _n+J_n\beta _n)\) for some \(J=(J_1,\ldots ,J_n)\in ({\mathbb {S}}_A)^n\). By hypothesis, we know that \(F_\emptyset (z),F_\ell (z)\in \mathbb {R}\). Since \(\ell \le h\) for all \(h\in H\), given any \(K\in \mathcal {P}(n)\) with \(K\subset H\), we have that:

  • \(e_\ell \cdot _\sigma e_K=e_{K\cup \{\ell \}}\) and \([J_{K\cup \{\ell \}},G_K(z)]=J_\ell [J_K,G_K(z)]\) if \(\ell \not \in K\),

  • \(e_\ell \cdot _\sigma e_K=-e_{K\setminus \{\ell \}}\) and, thanks to Artin’s theorem, \([J_{K\setminus \{\ell \}},G_K(z)]=-J_\ell [J_K,G_K(z)]\) if \(\ell \in K\).

In particular, it holds:

$$\begin{aligned} (f\cdot _\sigma g)(x)=\,&\textstyle \sum _{K\in \mathcal {P}(n),K\subset H}[J_K,F_\emptyset (z)G_K(z)]\\&+\textstyle \sum _{K\in \mathcal {P}(n),K\subset H,\ell \not \in K}[J_{K\cup \{\ell \}},F_\ell (z)G_K(z)]\\&\textstyle +\sum _{K\in \mathcal {P}(n),K\subset H,\ell \in K}(-[J_{K\setminus \{\ell \}},F_\ell (z)G_K(z)])\\ =\,&\textstyle F_\emptyset (z)\sum _{K\in \mathcal {P}(n),K\subset H}[J_K,G_K(z)]\\&\textstyle +J_\ell F_\ell (z)\sum _{K\in \mathcal {P}(n),K\subset H,\ell \not \in K}[J_K,G_K(z)]\\&\textstyle +J_\ell F_\ell (z)\sum _{K\in \mathcal {P}(n),K\subset H,\ell \in K}[J_K,G_K(z)]\\ =\,&\textstyle (F_\emptyset (z)+J_\ell F_\ell (z))(\sum _{K\in \mathcal {P}(n),K\subset H}[J_K,G_K(z)])=f(x)g(x), \end{aligned}$$

as desired. \(\square \)

Slice regular functions

Assumption 3.1

Throughout this section, we assume that D is open in \(\mathbb {C}^n\).

Complex structures on \(A \otimes \mathbb {R}^{2^n}\)

Let \(F=\sum _{K\in \mathcal {P}(n)}e_KF_K:D\rightarrow A\otimes \mathbb {R}^{2^n}\) be a \(\mathscr {C}^1\) function, i.e., suppose each \(F_K:D\rightarrow A\) is of class \(\mathscr {C}^1\) in the usual real sense. Given a family \(\mathcal {J}=\{\mathcal {J}_h\}_{h=1}^n\) consisting of n complex structures on \(A\otimes \mathbb {R}^{2^n}\), we say that a function F is holomorphic w.r.t. \(\mathcal {J}\) if, for all \(z\in D\), it holds:

$$\begin{aligned} \frac{\partial F}{\partial \alpha _h}(z)+\mathcal {J}_h\left( \frac{\partial F}{\partial \beta _h}(z)\right) =0\, \text { for all } h\in \{1,\ldots ,n\}, \end{aligned}$$
(43)

where \(z=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\) are the coordinates of \(\mathbb {C}^n\), \(\frac{\partial F}{\partial \alpha _h}=\sum _{K\in \mathcal {P}(n)}e_K\frac{\partial F_K}{\partial \alpha _h}\) and \(\frac{\partial F}{\partial \beta _h}=\sum _{K\in \mathcal {P}(n)}e_K\frac{\partial F_K}{\partial \beta _h}\). For short, in what follows, we will often denote \(\partial _{\alpha _h}\) and \(\partial _{\beta _h}\) the partial derivatives \(\frac{\partial }{\partial \alpha _h}\) and \(\frac{\partial }{\partial \beta _h}\), respectively.

We are interested in finding all the families \(\mathcal {J}=\{\mathcal {J}_h\}_{h=1}^n\) having the following two universal/algebraic properties:

Property 3.2

Each complex structure \(\mathcal {J}_h\) of \(A\otimes \mathbb {R}^{2^n}\) is the extension of a complex structure of \(\mathbb {R}^{2^n}\), which we call again \(\mathcal {J}_h\), via tensor product in the sense that \(\mathcal {J}_h(a\otimes x)=a \otimes \mathcal {J}_h(x)\,\) for all \(a\in A\) and \(x\in \mathbb {R}^{2^n}\). Equivalently, given any \(K\in \mathcal {P}(n)\) and \(a\in A\), if \(\mathcal {J}_h(e_K)=\sum _{H\in \mathcal {P}(n)}e_Hj^{\scriptscriptstyle (h,K)}_H\in \mathbb {R}^{2^n}\), then \(\mathcal {J}_h(e_Ka)=\sum _{H\in \mathcal {P}(n)}e_H(j^{\scriptscriptstyle (h,K)}_Ha)\in A\otimes \mathbb {R}^{2^n}\).

Property 3.3

All polynomial stem functions \(F:\mathbb {C}^n\rightarrow A\otimes \mathbb {R}^{2^n}\) are holomorphic w.r.t. \(\mathcal {J}\).

Note that, if we apply the latter property to the polynomial stem functions \(\{(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\mapsto \alpha _h+e_h\beta _h\}_{h=1}^n\), which induce the coordinate monomials \(x_1,\ldots ,x_n\), then we deduce that \(1+\mathcal {J}_h(e_h)=0\) for all \(h\in \{1,\ldots ,n\}\), which is equivalent to say that \(\mathcal {J}_h(1)=e_h\) or \(\mathcal {J}_h(e_h)=-1\).

Our next result shows that there exists a unique family \(\mathcal {J}\) with the mentioned two properties.

Proposition 3.4

There exists a unique family \(\{\mathcal {J}_h:A\otimes \mathbb {R}^{2^n}\rightarrow A\otimes \mathbb {R}^{2^n}\}_{h=1}^n\) of complex structures satisfying Properties 3.2 and 3.3. Each endomorphism \(\mathcal {J}_h\) is characterized by Property 3.2 and the following condition: for each \(K\in \mathcal {P}(n)\), it holds

$$\begin{aligned} \mathcal {J}_h(e_K)=(-1)^{|K \cap \{h\}|}e_{K \bigtriangleup \{h\}} \end{aligned}$$
(44)

or, equivalently,

$$\begin{aligned} \mathcal {J}_h(e_K)= \left\{ \begin{array}{ll} -e_{K \setminus \{h\}} &{} \quad \text { if }h\in K\\ e_{K \cup \{h\}} &{} \quad \text { if }h\not \in K \end{array} \right. . \end{aligned}$$
(45)

Proof

Let us prove (45) by induction on the cardinality |K| of \(K\in \mathcal {P}(n)\). If \(|K|=0\), then \(K=\emptyset \) and we just know that \(\mathcal {J}_h(e_\emptyset )=\mathcal {J}_h(1)=e_h\) for all \(h\in \{1,\ldots ,n\}\), so (45) is verified.

Suppose \(|K|\ge 1\). Let \(K=\{k_1,\ldots ,k_s\}\) with \(k_1<\ldots <k_s\) and let \(F:\mathbb {C}^n\rightarrow A\otimes \mathbb {R}^{2^n}\) be the monomial stem function inducing the monomial function \(x_K:=[(x_k)_{k\in K}]\). We have

$$\begin{aligned}\textstyle F(z)=\sum _{L\in \mathcal {P}(K)}e_L(\alpha _{K\setminus L}\beta _L), \end{aligned}$$

for all \(z=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\in \mathbb {C}^n\), where \(\mathcal {P}(K)\) denotes the set \(\{L\in \mathcal {P}(n):L\subset K\}\), \(\alpha _{K\setminus L}:=\prod _{h\in K\setminus L}\alpha _h\) and \(\beta _L:=\prod _{h\in L}\beta _h\), where \(\alpha _\emptyset =\beta _\emptyset :=1\). Choose \(h\in K\). It holds

$$\begin{aligned}&\textstyle \partial _{\alpha _h}F=\sum _{L\in \mathcal {P}(K),h\not \in L}e_L(\alpha _{K\setminus (L\cup \{h\})}\beta _L); \end{aligned}$$

moreover, thanks to Property 3.2 and the induction hypothesis, we have

$$\begin{aligned} \textstyle \mathcal {J}_h(\partial _{\beta _h}F)&\textstyle =\sum _{L\in \mathcal {P}(K),h\in L}\mathcal {J}_h(e_L)(\alpha _{K\setminus L}\beta _{L\setminus \{h\}})\\&\textstyle =\mathcal {J}_h(e_K)\beta _{K\setminus \{h\}}+\sum _{L\in \mathcal {P}(K)\setminus \{K\},h\in L}(-e_{L\setminus \{h\}})(\alpha _{K\setminus L}\beta _{L\setminus \{h\}})\\&\textstyle =\mathcal {J}_h(e_K)\beta _{K\setminus \{h\}}-\sum _{L\in \mathcal {P}(K),h\not \in L,L\ne K\setminus \{h\}}e_L(\alpha _{K\setminus (L\cup \{h\})}\beta _L)\\&\textstyle =\mathcal {J}_h(e_K)\beta _{K\setminus \{h\}}+e_{K\setminus \{h\}}\beta _{K\setminus \{h\}}-\partial _{\alpha _h}F. \end{aligned}$$

By Property 3.3, F is holomorphic w.r.t. \(\mathcal {J}\), so

$$\begin{aligned} \textstyle 0=\textstyle \partial _{\alpha _h}F+\mathcal {J}_h(\partial _{\beta _h}F)=\mathcal {J}_h(e_K)\beta _{K\setminus \{h\}}+e_{K\setminus \{h\}}\beta _{K\setminus \{h\}}. \end{aligned}$$

Consequently, \(\mathcal {J}_h(e_K)=-e_{K\setminus \{h\}}\) as desired. If \(h\not \in K\), then the preceding equality implies that \(\mathcal {J}_h(e_{K\cup \{h\}})=-e_K\), so \(\mathcal {J}_h(e_K)=\mathcal {J}_h(-\mathcal {J}_h(e_{K\cup \{h\}}))=e_{K\cup \{h\}}\). Equality (45) is proven.

It remains to show that the complex structures \(\mathcal {J}=\{\mathcal {J}_1,\ldots ,\mathcal {J}_n\}\) on \(A\otimes \mathbb {R}^{2^n}\) defined by Property 3.2 and equality (45) satisfies also Property 3.3. Equivalently, we have to prove that, given any \(\ell =(\ell _1,\ldots ,\ell _n)\in \mathbb {N}^n\), the monomial stem function F inducing \(x^\ell \) is holomorphic w.r.t. \(\mathcal {J}\). Given \(k\in \mathbb {N}\), let \(p_k,q_k\in \mathbb {R}[X,Y]\) be such that \((\alpha +i\beta )^k=p_k(\alpha ,\beta )+iq_k(\alpha ,\beta )\) for all \(\alpha ,\beta \in \mathbb {R}\), as in Definition 2.8. By the Cauchy-Riemann equations, we have that \(\partial _\alpha p_k=\partial _\beta q_k\) and \(\partial _\beta p_k=-\partial _\alpha q_k\). For each \(K=\{k_1,\ldots ,k_s\}\in \mathcal {P}(n)\setminus \{\emptyset \}\) with \(k_1<\ldots <k_s\), define the functions \(p_K,q_K:\mathbb {C}^n\rightarrow \mathbb {R}\) by \(p_K(z):=\prod _{k\in K}p_{\ell _k}(\alpha _k,\beta _k)\) and \(q_K(z):=\prod _{k\in K}q_{\ell _k}(\alpha _k,\beta _k)\), where \(z=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\in \mathbb {C}^n\). Define also \(p_\emptyset =q_\emptyset :\mathbb {C}^n\rightarrow \mathbb {R}\) as the function constantly equal to 1. Note that \(F=\sum _{K\in \mathcal {P}(n)}e_Kp_{\{1,\ldots ,n\}\setminus K}q_K\). Consequently, if \(h\in \{1,\ldots ,n\}\), it holds:

$$\begin{aligned} \partial _{\alpha _h}F=&\textstyle \sum _{K\in \mathcal {P}(n),h\in K}e_Kp_{\{1,\ldots ,n\}\setminus K}(\partial _{\alpha _h}q_{\ell _h})q_{K\setminus \{h\}}\\&\textstyle +\sum _{K\in \mathcal {P}(n),h\not \in K}e_K(\partial _{\alpha _h}p_{\ell _h})p_{\{1,\ldots ,n\}\setminus (K\cup \{h\})}q_K\\ =&\textstyle -\sum _{K\in \mathcal {P}(n),h\in K}e_Kp_{\{1,\ldots ,n\}\setminus K}(\partial _{\beta _h}p_{\ell _h})q_{K\setminus \{h\}}\\&\textstyle +\sum _{K\in \mathcal {P}(n),h\not \in K}e_K(\partial _{\beta _h}q_{\ell _h})p_{\{1,\ldots ,n\}\setminus (K\cup \{h\})}q_K\\ =&\textstyle -\sum _{K\in \mathcal {P}(n),h\not \in K}e_{K\cup \{h\}}p_{\{1,\ldots ,n\}\setminus (K\cup \{h\})}(\partial _{\beta _h}p_{\ell _h})q_K\\&\textstyle +\sum _{K\in \mathcal {P}(n),h\in K}e_{K\setminus \{h\}}(\partial _{\beta _h}q_{\ell _h})p_{\{1,\ldots ,n\}\setminus K}q_{K\setminus \{h\}} \end{aligned}$$

and

$$\begin{aligned} \textstyle \mathcal {J}_h\left( \partial _{\beta _h}F\right) =&\textstyle \sum _{K\in \mathcal {P}(n),h\in K}\mathcal {J}_h(e_K)p_{\{1,\ldots ,n\}\setminus K}(\partial _{\beta _h}q_{\ell _h})q_{K\setminus \{h\}}\\&\textstyle +\sum _{K\in \mathcal {P}(n),h\not \in K}\mathcal {J}_h(e_K)(\partial _{\beta _h}p_{\ell _h})p_{\{1,\ldots ,n\}\setminus (K\cup \{h\})}q_K\\ =&\textstyle -\sum _{K\in \mathcal {P}(n),h\in K}e_{K\setminus \{h\}}p_{\{1,\ldots ,n\}\setminus K}(\partial _{\beta _h}q_{\ell _h})q_{K\setminus \{h\}}\\&\textstyle +\sum _{K\in \mathcal {P}(n),h\not \in K}e_{K\cup \{h\}}(\partial _{\beta _h}p_{\ell _h})p_{\{1,\ldots ,n\}\setminus (K\cup \{h\})}q_K. \end{aligned}$$

Consequently, \(\partial _{\alpha _h}F+\mathcal {J}_h\left( \partial _{\beta _h}F\right) =0\), as desired. \(\square \)

The above complex structures \(\mathcal {J}_1,\ldots ,\mathcal {J}_n\) commute.

Lemma 3.5

The complex structures \(\mathcal {J}_1,\ldots ,\mathcal {J}_n\) on \(\mathbb {R}^{2^n}\) defined in Proposition 3.4 commute, that is \(\mathcal {J}_h\mathcal {J}_k=\mathcal {J}_k\mathcal {J}_h\) for all \(h,k\in \{1,\ldots ,n\}\).

Proof

Let \(K \in \mathcal {P}(n)\) and let \(h,k \in \{1,\ldots ,n\}\) with \(h\ne k\). We have

$$\begin{aligned} (\mathcal {J}_h\mathcal {J}_k)(e_K)= \left\{ \begin{array}{ll} -\mathcal {J}_h(e_{K \setminus \{k\}}) &{} \quad \text { if }k\in K \\ \mathcal {J}_h(e_{K \cup \{k\}}) &{} \quad \text { if }k\not \in K. \end{array} \right. \end{aligned}$$

Therefore

$$\begin{aligned} (\mathcal {J}_h\mathcal {J}_k)(e_K)= \left\{ \begin{array}{ll} e_{K \setminus \{h,k\}} &{} \quad \text { if }h,k \in K \\ -e_{(K \cup \{h\}) \setminus \{k\}} &{} \quad \text { if }h \notin K, k \in K \\ -e_{(K \cup \{k\}) \setminus \{h\}} &{} \quad \text { if }h \in K, k \notin K \\ e_{K \cup \{h,k\}} &{} \quad \text { if }h \notin K, k \notin K \end{array} \right. \end{aligned}$$

is symmetric in h and k. \(\square \)

Remark 3.6

The mentioned complex structures \(\mathcal {J}_1,\ldots ,\mathcal {J}_n\) on \(\mathbb {R}^{2^n}\) can also be characterized as the unique complex structures on \(\mathbb {R}^{2^n}\) satisfying the following two conditions:

  • \((\mathrm {i})\) \(\mathcal {J}_h\mathcal {J}_k=\mathcal {J}_k\mathcal {J}_h\) for all \(h,k\in \{1,\ldots ,n\}\).

  • \((\mathrm {ii})\) \(\mathcal {J}_h(e_K)=e_{K\cup \{h\}}\) for all \(h \in \{1,\ldots ,n\}\) and \(K \in \mathcal {P}(n)\) such that \(h<k\) for all \(k \in K\).

We have only to verify the uniqueness of \(\mathcal {J}_1,\ldots ,\mathcal {J}_n\). Let \(\mathcal {J}_1^{\prime },\ldots ,\mathcal {J}_n^{\prime }\) be complex structures on \(\mathbb {R}^{2^n}\) satisfying conditions \((\mathrm {i})\) and \((\mathrm {ii})\). Applying \((\mathrm {ii})\) with \(K=\emptyset \), we have that \(\mathcal {J}_h(1)=e_h\) and hence \(\mathcal {J}_h(e_h)=-1\) for all \(h\in \{1,\ldots ,n\}\). Let \(K=\{k_1,\ldots ,k_s\}\in \mathcal {P}(n)\setminus \{\emptyset \}\) with \(k_1<\ldots <k_s\), and let \(h\in \{1,\ldots ,n\}\). We have to prove that \(\mathcal {J}_h^{\prime }(e_K)=-e_{K \setminus \{h\}}\) if \(h \in K\) and \(\mathcal {J}_h^{\prime }(e_K)=e_{K \cup \{h\}}\) if \(h \not \in K\). First, suppose \(h\not \in K\). If \(h<k\) for all \(k\in K\), then we are done by \((\mathrm {ii})\). If \(h>k\) for some \(k\in K\), then there exists a unique \(t\in \{1,\ldots ,s\}\) such that \(k_t<h<k_{t+1}\), where \(k_{s+1}:=n+1\). Define \(K':=\{k_{t+1},\ldots ,k_s\}\) if \(t+1\le s\), and \(K':=\emptyset \) otherwise. By \((\mathrm {i})\) and \((\mathrm {ii})\), we have:

$$\begin{aligned} \mathcal {J}_h^{\prime }(e_K)&=(\mathcal {J}_h^{\prime }\mathcal {J}_{k_1}^{\prime }\cdots \mathcal {J}_{k_t}^{\prime })(e_{K^{\prime }})=(\mathcal {J}_{k_1}^{\prime }\cdots \mathcal {J}_{k_t}^{\prime })\big (\mathcal {J}_h^{\prime }(e_{K^{\prime }})\big )\\&=(\mathcal {J}_{k_1}^{\prime }\cdots \mathcal {J}_{k_t}^{\prime })(e_{K'\cup \{h\}})=e_{K\cup \{h\}}. \end{aligned}$$

Finally, if \(h\in K\), then \(\mathcal {J}_h(e_{K\setminus \{h\}})=e_K\), so \(\mathcal {J}_h(e_K)=\mathcal {J}_h(\mathcal {J}_h(e_{K\setminus \{h\}}))=-e_{K\setminus \{h\}}\). \(\square \)

Notation 3.7

In what follows, we denote \(\mathcal {J}=\{\mathcal {J}_1,\ldots ,\mathcal {J}_n\}\) the family of complex structures on \(A\otimes \mathbb {R}^{2^n}\) defined in Proposition 3.4.

Definition 3.8

Let \(F:D \rightarrow A \otimes \mathbb {R}^{2^n}\) be a \(\mathscr {C}^1\) stem function. For each \(h \in \{1,\ldots ,n\}\), we denote \(\partial _h\) and \({{\overline{\partial }}}_h\) the Cauchy-Riemann operators w.r.t. the complex structures i on D and \(\mathcal {J}_h\) on \(A \otimes \mathbb {R}^{2^n}\), that is

$$\begin{aligned} \partial _hF=\frac{1}{2}\left( \dfrac{\partial F}{\partial \alpha _h}-\mathcal {J}_h\left( \dfrac{\partial F}{\partial \beta _h}\right) \right) \,\,\text { and }\,\, \quad {{\overline{\partial }}}_hF=\frac{1}{2}\left( \dfrac{\partial F}{\partial \alpha _h}+\mathcal {J}_h\left( \dfrac{\partial F}{\partial \beta _h}\right) \right) , \end{aligned}$$

where \((\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\) are the coordinates of D. \(\square \)

As a consequence of Lemma 3.5, each operator of the type \(\partial _h\) or \({{\overline{\partial }}}_h\) commutes with each other:

$$\begin{aligned} \partial _h\partial _k=\partial _k\partial _h,\quad \partial _h{{\overline{\partial }}}_k={{\overline{\partial }}}_k\partial _h \,\,\text { and }\,\, {{\overline{\partial }}}_h{{\overline{\partial }}}_k={{\overline{\partial }}}_k{{\overline{\partial }}}_h \end{aligned}$$
(46)

for all \(h,k \in \{1,\ldots ,n\}\).

Lemma 3.9

Let \(F:D \rightarrow A \otimes \mathbb {R}^{2^n}\) be a \(\mathscr {C}^1\) stem function and let \(h \in \{1,\ldots ,n\}\). For each \(K \in \mathcal {P}(n)\), denote \((\partial _hF)_K\) and \(({{\overline{\partial }}}_hF)_K\) the K-components of \(\partial _hF\) and \({{\overline{\partial }}}_hF\), respectively. Then, for all \(K \in \mathcal {P}(n)\), it holds:

$$\begin{aligned} (\partial _hF)_K=\frac{1}{2}\left( \dfrac{\partial F_K}{\partial \alpha _h}+\dfrac{\partial F_{K \bigtriangleup \{h\}}}{\partial \beta _h}(-1)^{|K \cap \{h\}|}\right) \end{aligned}$$
(47)

and

$$\begin{aligned} ({{\overline{\partial }}}_hF)_K=\frac{1}{2}\left( \dfrac{\partial F_K}{\partial \alpha _h}-\dfrac{\partial F_{K \bigtriangleup \{h\}}}{\partial \beta _h}(-1)^{|K \cap \{h\}|}\right) . \end{aligned}$$
(48)

In particular, \(\partial _hF:D \rightarrow A \otimes \mathbb {R}^{2^n}\) and \({{\overline{\partial }}}_hF:D \rightarrow A \otimes \mathbb {R}^{2^n}\) are stem functions.

Proof

Let \(K \in \mathcal {P}(n)\). Equation (47) follows immediately from the following computation:

$$\begin{aligned} 2 \, \partial _hF=&\textstyle \sum _{K \in \mathcal {P}(n)} \big (e_K \partial _{\alpha _h}F_K-\mathcal {J}_h(e_K) \partial _{\beta _h}F_K\big )\\ =&\textstyle \sum _{K\in \mathcal {P}(n),h\in K} \big (e_K \partial _{\alpha _h}F_K+e_{K \setminus \{h\}}\partial _{\beta _h}F_K\big )\\&\textstyle +\sum _{K\in \mathcal {P}(n),h\not \in K} \big (e_K \partial _{\alpha _h}F_K-e_{K \cup \{h\}} \partial _{\beta _h}F_K\big )\\ =&\textstyle \sum _{K\in \mathcal {P}(n),h\in K} e_K\big (\partial _{\alpha _h}F_K- \partial _{\beta _h}F_{K \setminus \{h\}}\big )\\&+\textstyle \sum _{K\in \mathcal {P}(n),h\not \in K} e_K\big (\partial _{\alpha _h}F_K+ \partial _{\beta _h}F_{K\cup \{h\}}\big ). \end{aligned}$$

Similarly, we have that

$$\begin{aligned} \textstyle 2 \, {{\overline{\partial }}}_hF=&\sum _{K\in \mathcal {P}(n),h\in K} e_K\big (\partial _{\alpha _h}F_K+ \partial _{\beta _h}F_{K \setminus \{h\}}\big )\\&\textstyle +\sum _{K\in \mathcal {P}(n),h\not \in K} e_K\big (\partial _{\alpha _h}F_K- \partial _{\beta _h}F_{K\cup \{h\}}\big ). \end{aligned}$$

Consequently, (48) holds.

It remains to show that \(\partial _hF\) and \({{\overline{\partial }}}_hF\) are stem functions. Let \(j \in \{1,\ldots ,n\}\). Since F is a stem function, we know that \(F_K({\overline{z}}^j)=(-1)^{|K \cap \{j\}|}F_K(z)\) for all \(z \in D\). Fix \(z \in D\). Differentiating both members of the latter equality w.r.t. \(\alpha _h\) and \(\beta _h\), we obtain:

$$\begin{aligned} \partial _{\alpha _h}F_K({\overline{z}}^j)=(-1)^{|K \cap \{j\}|}\partial _{\alpha _h}F_K(z) \end{aligned}$$
(49)

and

$$\begin{aligned} \partial _{\beta _h}F_K({\overline{z}}^j)=(-1)^{|K \cap \{j\}|+|\{j\} \cap \{h\}|} \partial _{\beta _h}F_K(z). \end{aligned}$$

Since the integers \(|(K \bigtriangleup \{h\}) \cap \{j\}|+|\{j\} \cap \{h\}|\) and \(|K \cap \{j\}|\) have the same parity, the latter equality implies the next one:

$$\begin{aligned} \partial _{\beta _h}F_{K \bigtriangleup \{h\}}({\overline{z}}^j)=(-1)^{|K \cap \{j\}|} \partial _{\beta _h}F_{K\bigtriangleup \{h\}}(z). \end{aligned}$$
(50)

By combining (47), (49) and (50), we obtain that \((\partial _hF)_K({\overline{z}}^j)=(-1)^{|K \cap \{j\}|}(\partial _hF)_K(z)\) and hence \(\partial _hF\) is a stem function. Similarly, by using (48) instead of (47), we infer that \({{\overline{\partial }}}_hF\) is a stem function as well. \(\square \)

The last part of Lemma 3.9 allows to give the following definition.

Definition 3.10

Let \(F:D \rightarrow A\otimes \mathbb {R}^{2^n}\) be a \(\mathscr {C}^1\) stem function, let \(f=\mathcal {I}(F):\Omega _D \rightarrow A\) be the corresponding slice function and let \(h \in \{1,\ldots ,n\}\). We define the slice partial derivatives \(\frac{\partial f}{\partial x_h}\) and \(\frac{\partial f}{\partial x_h^c}\) of f as the following slice function in \(\mathcal {S}^0(\Omega _D,A)\):

$$\begin{aligned} \dfrac{\partial f}{\partial x_h}:=\mathcal {I}(\partial _hF) \,\,\text { and }\,\, \dfrac{\partial f}{\partial x_h^c}:=\mathcal {I}({{\overline{\partial }}}_hF). \, \end{aligned}$$
(51)

\(\square \)

Holomorphic stem functions, slice regular functions and polynomials

Let us introduce the concepts of holomorphic stem and slice regular functions. The reader will recall Assumptions 3.1 and 3.7, and Definition 3.8.

Definition 3.11

Let \(F:D \rightarrow A \otimes \mathbb {R}^{2^n}\) be a \(\mathscr {C}^1\) stem function. Given \(h \in \{1,\ldots ,n\}\), we say that F is h-holomorphic if \({{\overline{\partial }}}_hF=0\) on D. We call F holomorphic if it is h-holomorphic for all \(h \in \{1,\ldots ,n\}\). If F is holomorphic, then we say that \(\mathcal {I}(F):\Omega _D \rightarrow A\) is a slice regular function. We denote \(\mathcal {SR}(\Omega _D,A)\) the real vector subspace of \(\mathcal {S}(\Omega _D,A)\) of all slice regular functions. \(\square \)

Note that F is h-holomorphic if and only if, for each \(z=(z_1,\ldots ,z_n)\in D\), the restriction function \(F^{\scriptscriptstyle (z)}:(D_h(z),i)\rightarrow (A \otimes \mathbb {R}^{2^n},\mathcal {J}_h)\), sending w into \(F(z_1,\ldots ,z_{h-1},w,z_{h+1},\ldots ,z_n)\), is holomorphic in the usual sense. Here \(D_h(z)\) is the non-empty open subset of \(\mathbb {C}\) defined in (22).

A useful characterization of the h-holomorphicity of a stem function is as follows.

Lemma 3.12

Let \(F:D \rightarrow A \otimes \mathbb {R}^{2^n}\) be a \({\mathscr {C}}^1\) stem function with \(F=\sum _{K \in \mathcal {P}(n)}e_KF_K\) and let \(h \in \{1,\ldots ,n\}\). The following assertions are equivalent:

  • \((\mathrm {i})\) F is h-holomorphic.

  • \((\mathrm {ii})\) For each \(K \in \mathcal {P}(n)\) with \(h\not \in K\), the functions \(F_K\) and \(F_{K \cup \{h\}}\) satisfy the following Cauchy-Riemann equations

    $$\begin{aligned} \dfrac{\partial F_K}{\partial \alpha _h}=\dfrac{\partial F_{K \cup \{h\}}}{\partial \beta _h} \,\, \text { and } \,\, \dfrac{\partial F_K}{\partial \beta _h}=-\dfrac{\partial F_{K \cup \{h\}}}{\partial \alpha _h}. \end{aligned}$$
    (52)

Proof

By (48), we know that

$$\begin{aligned} \textstyle 2 \, {{\overline{\partial }}}_hF=\sum _{K\in \mathcal {P}(n),h\not \in K} \big (e_{K \cup \{h\}} (\partial _{\alpha _h}F_{K \cup \{h\}}+\partial _{\beta _h}F_K)+ e_K(\partial _{\alpha _h}F_K-\partial _{\beta _h}F_{K \cup \{h\}})\big ). \end{aligned}$$

As a consequence, \({{\overline{\partial }}}_hF=0\) if and only if (52) is satisfied. \(\square \)

By Definition 2.44, given any \(J\in {\mathbb {S}}_A\), we have that \(\Omega _D(J)=\Omega _D\cap (\mathbb {C}_J)^n\) and \(f_J:\Omega _D(J)\rightarrow A\) is the restriction of f to \(\Omega _D(J)\). By Assumptions 2.1 and 3.1, it follows that \(\Omega _D(J)\) is a non-empty open subset of \((\mathbb {C}_J)^n\).

The next result contains some characterizations of slice regularity.

Proposition 3.13

Let \(f\in \mathcal {S}^1(\Omega _D,A)\) and let \(F=\sum _{K \in \mathcal {P}(n)}e_KF_K:D \rightarrow A \otimes \mathbb {R}^{2^n}\) be the \({\mathscr {C}}^1\) stem function inducing f. The following assertions are equivalent:

  • \((\mathrm {i})\) f is slice regular.

  • \((\mathrm {i}')\) \(\displaystyle \frac{\partial f}{\partial x_h^c}=0\) on \(\Omega _D\) for all \(h\in \{1,\ldots ,n\}\).

  • \((\mathrm {ii})\) For each \(K \in \mathcal {P}(n)\) and for each \(h \in \{1,\ldots ,n\}\) with \(h\not \in K\), it holds:

    $$\begin{aligned} \dfrac{\partial F_K}{\partial \alpha _h}=\dfrac{\partial F_{K \cup \{h\}}}{\partial \beta _h} \,\,\text { and }\,\, \dfrac{\partial F_K}{\partial \beta _h}=-\dfrac{\partial F_{K \cup \{h\}}}{\partial \alpha _h}. \end{aligned}$$
  • \((\mathrm {iii})\) There exists \(J \in {\mathbb {S}}_A\) such that \(f_J:\Omega _D(J)\rightarrow A\) is holomorphic w.r.t. the complex structures on \(\Omega _D(J)\) and on A defined by the left multiplication by J; that is,

    $$\begin{aligned} \dfrac{\partial f_J}{\partial \alpha _h}(z)+J\dfrac{\partial f_J}{\partial \beta _h}(z)=0\,\text { for all } z\in \Omega _D(J)\text { and for all }h \in \{1,\ldots ,n\}, \end{aligned}$$
    (53)

    where \(z=(\alpha _1+J\beta _1,\ldots ,\alpha _n+J\beta _n)\) are the coordinates of points \(z\in (\mathbb {C}_J)^n\).

  • \((\mathrm {iii}^{\prime })\) For each \(J \in {\mathbb {S}}_A\), \(f_J\) is holomorphic in the sense of (53).

Proof

Equivalence \((\mathrm {i}) \Leftrightarrow (\mathrm {i}')\) is an immediate consequence of the fact that, by Proposition 2.12, a slice function is null if and only if its stem function is. Equivalence \((\mathrm {i}) \Leftrightarrow (\mathrm {ii})\) follows immediately from Lemma 3.12. The implication \((\mathrm {iii}^{\prime }) \Rightarrow (\mathrm {iii})\) is evident. We will show that \((\mathrm {ii})\Rightarrow (\mathrm {iii}^{\prime })\) and \((\mathrm {iii})\Rightarrow (\mathrm {ii})\) completing the proof.

Let \(J \in {\mathbb {S}}_A\), let \(x=(\alpha _1+J\beta _1,\ldots ,\alpha _n+J\beta _n) \in \Omega _D(J)\), let \(z=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n) \in D\) and let \(h \in \{1,\ldots ,n\}\). By Artin’s theorem, we have that

$$\begin{aligned} \textstyle f_J(x)=\sum _{H \in \mathcal {P}(n)}J^{|H|}F_H(z). \end{aligned}$$
(54)

Consequently, \(f_J(x)=\sum _{H\in \mathcal {P}(n),h\not \in H}J^{|H|}\big (F_H(z)+JF_{H \cup \{h\}}(z)\big )\) and hence

$$\begin{aligned} \left( \partial _{\alpha _h}f_J+J\partial _{\beta _h}f_J\right) (x) =&\textstyle \sum _{H\in \mathcal {P}(n),h\not \in H}J^{|H|} \big (\partial _{\alpha _h}F_H-\partial _{\beta _h}F_{H \cup \{h\}}\big )(z)\\&\textstyle +\sum _{H\in \mathcal {P}(n),h\not \in H}J^{|H|+1}\big (\partial _{\alpha _h}F_{H \cup \{h\}}+ \partial _{\beta _h}F_H\big )(z)=0. \end{aligned}$$

This proves implication \((\mathrm {ii})\Rightarrow (\mathrm {iii}^{\prime })\). Finally, suppose that \((\mathrm {iii})\) holds. Let us prove \((\mathrm {ii})\). Let \(z \in D\) and \(x \in \Omega _D\) be as above and let \(L \in \mathcal {P}(n)\). Recall that \({\overline{z}}^L \in D\) and \(x^{\, c,L} \in \Omega _D\). For each \(H \in \mathcal {P}(n)\), define the function \(F_{H,L}:D \rightarrow A\) by setting \(F_{H,L}(z):=F_H({\overline{z}}^L)\). Thanks to (4) and (54), we obtain that

$$\begin{aligned} \textstyle f_J(x)=\sum _{H \in \mathcal {P}(n)}J^{|H|}(-1)^{|H \cap L|}F_{H,L}(z). \end{aligned}$$

On the other hand, it is immediate to verify that \(\partial _{\alpha _h}F_{H,L}(z)=\partial _{\alpha _h}F_H({\overline{z}}^L)\) and \(\partial _{\beta _h}F_{H,L}(z)=(-1)^{|L \cap \{h\}|}\partial _{\beta _h}F_H({\overline{z}}^L)\). Using again Artin’s theorem, it follows that

$$\begin{aligned} 0=&\left( \partial _{\alpha _h}f_J+J\partial _{\beta _h}f_J\right) (x^{\, c,L})\\ =&\textstyle \sum _{H \in \mathcal {P}(n)}J^{|H|}(-1)^{|H \cap L|}\left( \partial _{\alpha _h}F_{H,L}+J\partial _{\beta _h}F_{H,L}\right) ({\overline{z}}^L)\\ =&\textstyle \sum _{H \in \mathcal {P}(n)}J^{|H|}(-1)^{|H \cap L|}\partial _{\alpha _h}F_H(z)\\&\textstyle +\sum _{H \in \mathcal {P}(n)}J^{|H|+1}(-1)^{|H \cap L|+|L \cap \{h\}|}\partial _{\beta _h}F_H(z). \end{aligned}$$

We have just proven that

$$\begin{aligned} 0=&\textstyle \sum _{H \in \mathcal {P}(n)}J^{|H|}(-1)^{|H \cap L|}\partial _{\alpha _h}F_H(z)\nonumber \\&+\sum _{H \in \mathcal {P}(n)}J^{|H|+1}(-1)^{|H \cap L|+|L \cap \{h\}|}\partial _{\beta _h}F_H(z). \end{aligned}$$
(55)

Fix \(K \in \mathcal {P}(n)\) and \(h \in \{1,\ldots ,n\}\) with \(h\not \in K\). Multiply both members of (55) by \((-1)^{|L \cap K|}\) and sum over all \(L \in \mathcal {P}(n)\). Bearing in mind Lemma 2.11, we obtain

$$\begin{aligned} 0=&\textstyle \sum _{H \in \mathcal {P}(n)}J^{|H|}\partial _{\alpha _h}F_H(z)\left( \sum _{L \in \mathcal {P}(n)}(-1)^{|H \cap L|+|L \cap K|}\right) \\&\textstyle +\sum _{H \in \mathcal {P}(n)}J^{|H|+1}\partial _{\beta _h}F_H(z)\left( \sum _{L \in \mathcal {P}(n)}(-1)^{|H \cap L|+|L \cap \{h\}|+|L \cap K|}\right) \\ =&\, \textstyle 2^nJ^{|K|}\partial _{\alpha _h}F_K(z)\\&\textstyle +\sum _{H \in \mathcal {P}(n)}J^{|H|+1}\partial _{\beta _h}F_H(z)\left( \sum _{L \in \mathcal {P}(n)}(-1)^{|H \cap L|+|L \cap (K \cup \{h\})|}\right) \\ =&\, \textstyle 2^nJ^{|K|}\partial _{\alpha _h}F_K(z)\\&\textstyle +2^nJ^{|K|+2}\partial _{\beta _h}F_{K \cup \{h\}}(z)=2^nJ^{|K|}\left( \partial _{\alpha _h}F_K(z)-\partial _{\beta _h}F_{K \cup \{h\}}(z)\right) , \end{aligned}$$

and hence \(\partial _{\alpha _h}F_K(z)=\partial _{\beta _h}F_{K \cup \{h\}}(z)\). Similarly, multiplying both members of (55) by \((-1)^{|L \cap (K \cup \{h\})|}\) and summing over all \(L \in \mathcal {P}(n)\), we obtain

$$\begin{aligned} 0=2^nJ^{|K|+1}\left( \partial _{\alpha _h}F_{K \cup \{h\}}(z)+\partial _{\beta _h}F_K(z)\right) , \end{aligned}$$

and hence \(\partial _{\beta _h}F_K(z)=-\partial _{\alpha _h}F_{K \cup \{h\}}(z)\). This proves \((\mathrm {ii})\). \(\square \)

Slice regular functions include polynomial functions.

Proposition 3.14

All polynomial functions from \(\Omega _D\) to A are slice regular.

Proof

For each \(k\in \{1,\ldots ,n\}\), it is immediate to see that the coordinate function \(x_k:\Omega _D\rightarrow A\) is slice regular; indeed \(\frac{\partial x_k}{\partial x_h^c}=0\) on \(\Omega _D\) for all \(h\in \{1,\ldots ,n\}\). Combining Lemma 2.43, Eq. (73) and equivalence \((\mathrm {i})\Leftrightarrow (\mathrm {i}')\) of Proposition 3.13, we deduce at once that all polynomials functions from \(\Omega _D\) to A are slice regular. \(\square \)

On the zeros of polynomials

This section deals with the zero set of polynomials over A in the case A is the division algebra \({\mathbb {H}}\) of quaternions, the one \(\mathbb {O}\) of octonions or the Clifford algebra \(\mathbb {R}_m= C \ell (0,m)\) for \(m\ge 3\). We apply a Fundamental Theorem of Algebra which is available on these alternative algebras (see [20, 18, 22] and [9,  Examples 9(1)]) to the polynomial in the first variable \(x_1\) which is obtained fixing the other variables and then we use arguments from real algebraic geometry to deduce the possible real dimensions of the zero set.

Given any function \(f:(Q_A)^n\rightarrow A\), we denote V(f) its zero set, i.e.

$$\begin{aligned} V(f):=\{x\in (Q_A)^n:f(x)=0\}. \end{aligned}$$

If \(A={\mathbb {H}}\) and f is a polynomial in the sense of Definition 2.7, then the next result gives some properties of V(f). Identify \({\mathbb {H}}^n\) with \(\mathbb {R}^{4n}\) by choosing one of its real vector basis.

Proposition 3.15

Let \(f:{\mathbb {H}}^n\rightarrow {\mathbb {H}}\) be a nonconstant polynomial. Then V(f) is a nonempty real algebraic subset of \(\mathbb {R}^{4n}\) and its dimension \(\dim _\mathbb {R}(V(f))\), as a real algebraic set, satisfies the following estimates:

$$\begin{aligned} 4n-4\le \dim _\mathbb {R}(V(f))\le 4n-2. \end{aligned}$$
(56)

Proof

The equation \(f=0\) in n quaternionic variables is equivalent to a system of four real polynomial equations in 4n real variables, so V(f) is a real algebraic subset of \(\mathbb {R}^{4n}\).

Let us prove (56) by induction on \(n\ge 1\). If \(n=1\), this is an immediate consequence of the Fundamental Theorem of Algebra for quaternions [20], FTA for short. Let \(n\ge 2\) and let \(f:{\mathbb {H}}^n\rightarrow {\mathbb {H}}\) be a nonconstant polynomial function. We can assume that f has the following form: \(f(x)=\sum _{h=0}^dx_1^hp_h(x')\), where \(d\ge 0\), each \(p_h:{\mathbb {H}}^{n-1}\rightarrow {\mathbb {H}}\) is a polynomial function in the variables \(x':=(x_2,\ldots ,x_n)\) and \(p_d\) does not vanish identically on \({\mathbb {H}}^{n-1}\). Note that the zero set W of \(p_d\) in \({\mathbb {H}}^{n-1}\) is either empty if \(p_d\) is constant or, by induction, \(\dim _\mathbb {R}(W)\le 4(n-1)-2=4n-6\) if \(p_d\) is not constant; in particular, \(W\ne {\mathbb {H}}^{n-1}\).

Assume that \(d\ge 1\) and let \(y'\not \in W\). Then the FTA implies that \(V(f)\cap ({\mathbb {H}}\times \{y'\})\) is either a nonempty finite set F or a nonempty finite union S of 2-spheres of \({\mathbb {H}}=\mathbb {R}^4\) or a nonempty set of the form \(F\cup S\). In particular, \(V(f)\ne \emptyset \). Let \(\pi :V(f)\rightarrow {\mathbb {H}}^{n-1}\) be the projection \(\pi (x_1,x'):=x'\). We have just proven that \(\pi ^{-1}(y')\) is nonempty and \(\dim _\mathbb {R}(\pi ^{-1}(y'))\le 2\) for all \(y'\not \in W\). Evidently, if \(y'\in W\), then \(\pi ^{-1}(y')\subset {\mathbb {H}}\times \{y'\}\) and hence \(\dim _\mathbb {R}(\pi ^{-1}(y'))\le 4\). By the version of Sard’s theorem in real algebraic geometry, we deduce at once that

$$\begin{aligned} 4n-4=\dim _\mathbb {R}({\mathbb {H}}^{n-1})\le \dim _\mathbb {R}(\pi ^{-1}({\mathbb {H}}^{n-1}\setminus W))\le \dim _\mathbb {R}({\mathbb {H}}^{n-1})+2=4n-2 \end{aligned}$$

and

$$\begin{aligned} \dim _\mathbb {R}(\pi ^{-1}(W))\le \dim _\mathbb {R}(W)+4\le (4n-6)+4=4n-2. \end{aligned}$$

Since \(\dim _\mathbb {R}(V(f))=\max \{\dim _\mathbb {R}(\pi ^{-1}({\mathbb {H}}^{n-1}\setminus W)),\dim _\mathbb {R}(\pi ^{-1}(W))\}\), we are done.

Assume now that \(d=0\). In this case it must be \(n>1\), since f is nonconstant, and f coincides with the polynomial \(p_0(x')\) in the variables \(x_2,\ldots , x_n\). From the previous argument applied to \(p_0\), we obtain that V(f) is nonempty and \(4(n-1)-4\le \dim _\mathbb {R}(V(p_0)\cap {\mathbb {H}}^{n-1})\le 4(n-1)-2\), from which it follows again that

$$\begin{aligned} 4(n-1)\le \dim _\mathbb {R}(V(f))\le 4(n-1)+2. \end{aligned}$$

\(\square \)

Example 3.16

Estimates (56) are sharp. Suppose that \(n\ge 2\). Consider the polynomial functions \(f_1,f_2,f_3:{\mathbb {H}}^n\rightarrow {\mathbb {H}}\) defined by setting

$$\begin{aligned} f_1(x):=x_1, \qquad f_2(x):=x_1^2+x_2^2+1, \qquad f_3(x):=x_1^2+1, \end{aligned}$$

for all \(x=(x_1,x_2,\ldots ,x_n)\in {\mathbb {H}}^n\). Evidently, \(\dim _\mathbb {R}(V(f_1))=4n-4\) and \(\dim _\mathbb {R}(V(f_3))=4n-2\). Let us study \(V(f_2)\). Note that \(x_2^2+1\) is a positive real number if and only if \(x_2\) belongs to the 3-dimensional semi-algebraic subset S of \({\mathbb {H}}\) defined by

$$\begin{aligned} S:=\mathbb {R}\cup \{x_2\in {\mathbb {H}}:\mathrm {Re}(x_2)=0,n(x_2)<1\} \end{aligned}$$

Let \(\pi :V(f_2)\rightarrow {\mathbb {H}}^{n-1}\) be the projection \(\pi (x,x'):=x'\), where \(x':=(x_2,\ldots ,x_n)\). Note that \(\pi ^{-1}(y')\) consists of a single point if \(y'\in {\mathbb {S}}_A\times {\mathbb {H}}^{n-2}\), a 2-sphere if \(y'\in S\times {\mathbb {H}}^{n-2}\) and two distinct points if \(y'\in {\mathbb {H}}^{n-1}\setminus (({\mathbb {S}}_A\cup S)\times {\mathbb {H}}^{n-2})\). As a consequence, we have:

  • \(\dim _\mathbb {R}(\pi ^{-1}({\mathbb {S}}_A\times {\mathbb {H}}^{n-2}))=\dim _\mathbb {R}({\mathbb {S}}_A\times {\mathbb {H}}^{n-2})+0=2+4(n-2)=4n-6\),

  • \(\dim _\mathbb {R}(\pi ^{-1}(S\times {\mathbb {H}}^{n-2}))=\dim _\mathbb {R}(S\times {\mathbb {H}}^{n-2})+2=3+4(n-2)+2=4n-3\),

  • \(\dim _\mathbb {R}(\pi ^{-1}({\mathbb {H}}^{n-1}\setminus (({\mathbb {S}}_A\cup S)\times {\mathbb {H}}^{n-2})))=\dim _\mathbb {R}({\mathbb {H}}^{n-1})+0=4(n-1)=4n-4\).

It follows that \(\dim _\mathbb {R}(V(f_2))=\max \{4n-6,4n-3,4n-4\}=4n-3\). \(\square \)

Proposition 3.15 remains valid over the octonion algebra \(\mathbb {O}\). The proof is identical, thanks to the fact that the Fundamental Theorem of Algebra still holds in this case (see [18, 22]).

Proposition 3.17

Let \(f:\mathbb {O}^n\rightarrow \mathbb {O}\) be a nonconstant polynomial. Then V(f) is a nonempty real algebraic subset of \(\mathbb {R}^{8n}\) and it holds

$$\begin{aligned} 8n-8\le \dim _\mathbb {R}(V(f))\le 8n-2. \end{aligned}$$
(57)

Remark 3.18

Suppose that \(n\ge 2\). Let \(f_1,f_2,f_3:\mathbb {O}^n\rightarrow \mathbb {O}\) be polynomial functions defined as in Example 3.16. Repeating the same considerations we made in the mentioned example, we obtain that \(\dim _\mathbb {R}(V(f_1))=8n-8\), \(\dim _\mathbb {R}(V(f_2))=8n-3\) and \(\dim _\mathbb {R}(V(f_3))=8n-2\).

\(\square \)

A weaker version of Proposition 3.15 is valid also over all Clifford algebra \(\mathbb {R}_m\) with \(m\ge 3\).

Choose \(m\ge 3\) and equip \(\mathbb {R}_m\) with the Clifford conjugation. Given any \(a\in \mathbb {R}_m\), we write \(a=\sum _{K\in \mathcal {P}(m)}a_Ke_K\) for \(a_K\in \mathbb {R}\). Recall that \(t(a)=2\sum '_Ka_Ke_K\) and \(n(a)=\sum '_K\langle a,ae_K\rangle e_K\), where \(\sum '_K=\sum _{K\in \mathcal {P}(m),|K|\equiv 0,3\,(\text {mod } 4)}\) and \(\langle \cdot ,\cdot \rangle \) is the standard Euclidean scalar product on \(\mathbb {R}_m=\mathbb {R}^{2^m}\), see Sect. 3.2 of [17]. As a consequence, \(a=\sum _{K\in \mathcal {P}(m)}a_Ke_K\) belongs to \(Q_{\mathbb {R}_m}\) if and only if \(a_K=0\) and \(\langle a,ae_K\rangle =0\) for all \(K\in \mathcal {P}(m)\setminus \{\emptyset \}\) such that \(|K|\equiv 0,3\,(\text {mod }4)\). Moreover, \(a=\sum _{K\in \mathcal {P}(m)}a_Ke_K\) belongs to \({\mathbb {S}}_{\mathbb {R}_m}\) if and only if \(a\in Q_{\mathbb {R}_m}\), \(a_\emptyset =0\) and \(\langle a,a\rangle =1\). It turns out that \(Q_{\mathbb {R}_m}\) and \({\mathbb {S}}_{\mathbb {R}_m}\) are real algebraic subsets of \(\mathbb {R}_m=\mathbb {R}^{2^m}\). Since \(Q_{\mathbb {R}_m}\setminus \mathbb {R}\) is semi-algebraically homeomorphic to \({\mathbb {S}}_{\mathbb {R}_m}\times \{(\alpha ,\beta )\in \mathbb {R}^2:\beta >0\}\), we have that

$$\begin{aligned} \dim _\mathbb {R}({\mathbb {S}}_{\mathbb {R}_m})=\dim _\mathbb {R}(Q_{\mathbb {R}_m})-2. \end{aligned}$$
(58)

Note that \(Q_{\mathbb {R}_3}\subset Q_{\mathbb {R}_m}\); moreover, \(\dim _\mathbb {R}(Q_{\mathbb {R}_3})=6\), see Examples 1(3) of [8] and [13,  Example 1.15]. It follows that \(\dim _\mathbb {R}(Q_{\mathbb {R}_m})\ge 6\). Denote \(\mathbb {R}^{m+1}\) the real vector subspace of \(\mathbb {R}_m\) of paravectors \(a=a_0+\sum _{h=1}^ma_he_h\), where \(a_0:=a_\emptyset \) and \(a_h:=a_{\{h\}}\) (and \(e_h=e_{\{h\}}\)). It is easy to see that \(\mathbb {R}^{m+1}\) is contained in \(Q_{\mathbb {R}_m}\).

The Fundamental Theorem of Algebra still holds for one variable polynomials with paravector Clifford coefficients (see [9,  Examples 9(1)]). This fact allows to generalize the first statement of Proposition 3.15 to this setting.

Proposition 3.19

Let \(f:(Q_{\mathbb {R}_m})^n\rightarrow \mathbb {R}_m\) be a nonconstant polynomial with paravector coefficients. Then V(f) is a nonempty real algebraic subset of \((Q_{\mathbb {R}_m})^n\subset \mathbb {R}^{n2^m}\).

Proof

We proceed by induction on n. If \(n=1\), the FTA [9,  Examples 9(1)] implies the thesis. Assume \(n\ge 2\) and write \(f(x)=\sum _{h=0}^dx_1^hp_h(x')\), where \(d\ge 0\), each \(p_h:(Q_{\mathbb {R}_m})^{n-1}\rightarrow \mathbb {R}_m\) is a polynomial function in the variables \(x':=(x_2,\ldots ,x_n)\) and \(p_d\) does not vanish identically on \((Q_{\mathbb {R}_m})^{n-1}\). If \(d=0\), then \(f(x)=p_0(x')\) is nonconstant and therefore \(V(f)\ne \emptyset \) by the induction assumption. If \(d\ge 1\), then for every fixed point \(y'=(t_2,\ldots ,t_n)\in \mathbb {R}\times \cdots \times \mathbb {R}\subset (Q_{\mathbb {R}_m})^{n-1}\), the function \(f(x_1,y')=\sum _{h=0}^dx_1^hp_h(y')\) is a nonconstant polynomial in the variable \(x_1\) with paravector coefficients. Therefore V(f) is nonempty. \(\square \)

When \(n>1\), the zero set V(f) is infinite for every nonconstant polynomial with paravector coefficients. It is difficult to obtain general estimates for the real dimension of V(f). The next proposition suggests the types of results one can expect.

Proposition 3.20

Let \(f:Q_{\mathbb {R}_m}\times (\mathbb {R}^{m+1})^{n-1}\rightarrow \mathbb {R}_m\) be a function of the form

$$\begin{aligned} \textstyle f(x_1,x')=x_1^d+\sum _{h=0}^{d-1}x_1^hp_h(x') \end{aligned}$$
(59)

for all \((x_1,x')\in Q_{\mathbb {R}_m}\times (\mathbb {R}^{m+1})^{n-1}\), where \(d\ge 1\) and each \(p_h:(Q_{\mathbb {R}_m})^{n-1}\rightarrow \mathbb {R}_m\) is a polynomial function in the variables \(x'=(x_2,\ldots ,x_n)\) such that

$$\begin{aligned} p_h((\mathbb {R}^{m+1})^{n-1})\subset \mathbb {R}^{m+1}\,\text { for all }h\in \{0,\ldots ,d-1\}; \end{aligned}$$
(60)

in the case \(n=1\), (60) means that the \(p_h\)’s are elements of \(\mathbb {R}^{m+1}\). Then \(f^{-1}(0)\) is a nonempty real algebraic subset of \(\mathbb {R}^{2^m+(m+1)(n-1)}\) contained in \(Q_{\mathbb {R}_m}\times (\mathbb {R}^{m+1})^{n-1}\), and it holds

$$\begin{aligned} (m+1)(n-1)\le \dim _\mathbb {R}(f^{-1}(0))\le (m+1)(n-1)+\dim _\mathbb {R}(Q_{\mathbb {R}_m})-2. \end{aligned}$$
(61)

Proof

If \(n=1\), the statement follows immediately from (58), (60) and Examples 9(1) of [9]. Suppose that \(n\ge 2\). Let \(y'\in (\mathbb {R}^{m+1})^{n-1}\). Using (58), (60) and Examples 9(1) of [9] again, we have that \(f^{-1}(0)\cap (Q_{\mathbb {R}_m}\times \{y'\})\) is either a nonempty finite set F or a nonempty finite union S of ‘spheres’ semi-algebraically homeomorphic to \({\mathbb {S}}_{\mathbb {R}_m}\) or a nonempty set of the form \(F\cup S\). In particular, \(f^{-1}(0)\ne \emptyset \) and, if \(\pi :f^{-1}(0)\rightarrow (\mathbb {R}^{m+1})^{n-1}\) is the projection \(\pi (x_1,x'):=x'\), then the fibers of \(\pi \) are nonempty real algebraic sets of dimension ranging from 0 to \(\dim _\mathbb {R}({\mathbb {S}}_{\mathbb {R}_m})\). This fact and (58) completes the proof. \(\square \)

Example 3.21

Suppose that \(n\ge 2\). Choose \(m\ge 3\). Set \(N:=(m+1)(n-1)\) and \(M:=\dim _\mathbb {R}(Q_{\mathbb {R}_m})\). Define the functions \(f_1,f_2,f_3:Q_{\mathbb {R}_m}\times (\mathbb {R}^{m+1})^{n-1}\rightarrow \mathbb {R}_m\) as in Example 3.16. Note that each of these functions satisfies (59) and (60). Evidently, \(\dim _\mathbb {R}(f_1^{-1}(0))=N\) and \(\dim _\mathbb {R}(f_3^{-1}(0))=N+M-2\). Let us study \(f_2^{-1}(0)\) following the strategy used in Example 3.16 to investigate \(V(f_2)\). Note that, if \(x_2\in \mathbb {R}^{m+1}\), then \(x_2^2+1\) is a positive real number if and only if \(x_2\in S\), where S is the m-dimensional semi-algebraic set defined by

$$\begin{aligned}\textstyle S:=\mathbb {R}\cup \big \{\sum _{h=0}^ma_he_h\in \mathbb {R}^{m+1}:a_0=0,\,\sum _{h=1}^ma_h^2<1\big \}. \end{aligned}$$

Let \(\partial S\) be the \((m-1)\)-sphere \(\{\sum _{h=0}^ma_he_h\in \mathbb {R}^{m+1}:a_0=0,\,\sum _{h=1}^ma_h^2=1\}\) and let \(\pi :f_2^{-1}(0)\rightarrow (\mathbb {R}^{m+1})^{n-1}\) be the projection \(\pi (x,x'):=x'\), where \(x':=(x_2,\ldots ,x_n)\). Note that \(\pi ^{-1}(y')\) consists of a single point if \(y'\in \partial S\times (\mathbb {R}^{m+1})^{n-2}\), a ‘sphere’ semi-algebraically homeomorphic to \({\mathbb {S}}_{\mathbb {R}_m}\) if \(y'\in S\times (\mathbb {R}^{m+1})^{n-2}\) and two distinct points if \(y'\in (\mathbb {R}^{m+1})^{n-1}\setminus ((S\cup \partial S)\times (\mathbb {R}^{m+1})^{n-2})\). As a consequence, we have:

  • \(\dim _\mathbb {R}(\pi ^{-1}(\partial S\times (\mathbb {R}^{m+1})^{n-2}))=(m-1)+(m+1)(n-2)<N\),

  • \(\dim _\mathbb {R}(\pi ^{-1}(S\times (\mathbb {R}^{m+1})^{n-2}))=m+(m+1)(n-2)+\dim _\mathbb {R}({\mathbb {S}}_{\mathbb {R}_m})=N-1+\dim _\mathbb {R}({\mathbb {S}}_{\mathbb {R}_m})\),

  • \(\dim _\mathbb {R}(\pi ^{-1}((\mathbb {R}^{m+1})^{n-1}\setminus ((S\cup \partial S)\times (\mathbb {R}^{m+1})^{n-2})))=N\).

Thanks to (58) and the fact that \(M=\dim _\mathbb {R}(Q_{\mathbb {R}_m})\ge 6\), it follows that

$$\begin{aligned} N-1+\dim _\mathbb {R}({\mathbb {S}}_{\mathbb {R}_m})=N-1+M-2=N+M-3\ge N+3>N. \end{aligned}$$

As a consequence, we have that \(\dim _\mathbb {R}(f_2^{-1}(0))=\dim _\mathbb {R}(\pi ^{-1}(S\times (\mathbb {R}^{m+1})^{n-2}))=N+M-3\). \(\square \)

One variable interpretation of slice regularity

We now show that the condition of slice regularity in several variables has an interpretation in terms of slice regularity in one variable. More precisely, we show that the slice regularity of a slice function \(f:\Omega _D\rightarrow A\) is equivalent to the slice regularity of all its \(2^n-1\) truncated spherical derivatives \(\mathcal {D}_\epsilon f\) w.r.t. the single variable \(\mathrm {x}_h\), where \(h-1\) is the order of \(\mathcal {D}_\epsilon \). The notion of truncated spherical \(\epsilon \)-derivatives was introduced in above Definition 2.24.

Let \(g:\Omega _D\rightarrow A\) be a function and let \(h\in \{1,\ldots ,n\}\). Recall that, by Definition 2.22, g is a slice function w.r.t. \(\mathrm {x}_h\) if, for each \(y=(y_1,\ldots ,y_n)\in \Omega _D\), the restriction function \(g_h^{\scriptscriptstyle (y)}:\Omega _{D,h}(y)\rightarrow A\), sending \(x_h\) into \(g_h^{\scriptscriptstyle (y)}(x_h):=g(y_1,\ldots ,y_{h-1},x_h,y_{h+1},\ldots ,y_n)\), is a slice function. Let us specialize this definition to the slice regular case.

Definition 3.22

Let \(g:\Omega _D\rightarrow A\) be a function and let \(h\in \{1,\ldots ,n\}\). We say that g is a slice regular function w.r.t. \(\mathrm {x}_h\) if, for each \(y\in \Omega _D\), the function \(g_h^{\scriptscriptstyle (y)}:\Omega _{D,h}(y)\rightarrow A\) is a slice regular function. \(\square \)

Theorem 3.23

Assume that \(n\ge 2\). Let \(f:\Omega _D\rightarrow A\) be a slice function. Then f is slice regular if and only if f is a slice regular function w.r.t. \(\mathrm {x}_1\) and, for each \(h\in \{2,\ldots ,n\}\) and each function \(\epsilon :\{1,\ldots ,h-1\}\rightarrow \{0,1\}\), the truncated spherical \(\epsilon \)-derivative \(\mathcal {D}_\epsilon f\) of f is a slice regular function w.r.t. \(\mathrm {x}_h\).

Proof

We use the notation introduced in (22) and (23). If \(x=(x_1,\ldots ,x_n)\in \Omega _D\), then \(\Omega _{D,h}(x)\) is the subset of \(Q_A\) defined by \(\Omega _{D,h}(x)=\{a\in A:(x_1,\ldots ,x_{h-1},a,x_{h+1},\ldots ,x_n)\in \Omega _D\}\). It holds \(\Omega _{D,h}(x)=\Omega _{D_h(z)}\), where \(D_h(z)=\{w\in \mathbb {C}:(z_1,\ldots ,z_{h-1},w,z_{h+1},\ldots ,z_n)\in D\}\).

Assume that f is slice regular. Let \(F=\sum _{H\in \mathcal {P}(n)}e_HF_H:D\rightarrow A\otimes \mathbb {R}^{2^n}\) be the stem function inducing f, let \(y=(y_1,\ldots ,y_n)=(\alpha _1+I_1\beta _1,\ldots ,\alpha _n+I_n\beta _n)\in \Omega _D\), let \(I:=(I_1,\ldots ,I_n)\) and let \(w=(w_1,\ldots ,w_n):=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\in D\). Let us prove that f is slice regular w.r.t. \(\mathrm {x}_1\). As seen in formula (27), it holds \(f(x_1,y')=F_1(z_1)+J_1F_2(z_1)\), where \(x_1=\alpha _1+J_1\beta _1\in \Omega _{D,1}(y)\), \(y'=(y_2,\ldots ,y_n)\), \(z_1=\alpha _1+i\beta _1\in D_1(w)\), and

$$\begin{aligned} F_1(z_1)\textstyle =\sum _{H\in \mathcal {P}(n),1\not \in H}[J_H,F_H(z_1,w')] \end{aligned}$$

and

$$\begin{aligned} F_2(z_1)\textstyle =\sum _{H\in \mathcal {P}(n),1\not \in H}[J_H,F_{H\cup \{1\}}(z_1,w')], \end{aligned}$$

with \(w'=(w_2,\ldots ,w_n)\) and \(J=(J_1,I_2,\ldots ,I_n)\). From implication \((\mathrm {i})\Rightarrow (\mathrm {ii})\) of Proposition 3.13, it follows that \(\partial _{\alpha _1}F_1(z_1)=\partial _{\beta _1}F_2(z_1)\) and \(\partial _{\beta _1}F_1(z_1)=-\partial _{\alpha _1}F_2(z_1)\), that is f is slice regular w.r.t. \(\mathrm {x}_1\) on \(\Omega _{D,1}(y)\).

Now let \(h\in \{2,\ldots ,n\}\) and let \(\epsilon :\{1,\ldots ,h-1\}\rightarrow \{0,1\}\) be any function. Set \(K_{h-1}:=\epsilon ^{-1}(1)\). Let us prove that \(\mathcal {D}_\epsilon f\) is slice regular w.r.t. \(\mathrm {x}_h\). From formula (28), it follows that \(\mathcal {D}_\epsilon f(y'',x_{h},{\hat{y}})=F_{h,1}(z_h)+J_hF_{h,2}(z_h)\), where

$$\begin{aligned} F_{h,1}(z_h)&\textstyle :=(\beta _{K_{h-1}})^{-1}\sum _{H\in \mathcal {P}(n),H_{h}=\emptyset }[L_H,F_{H\cup K_{h-1}}(z'',z_{h},{\hat{z}})],\\ F_{h,2}(z_h)&\textstyle :=(\beta _{K_{h-1}})^{-1}\sum _{H\in \mathcal {P}(n),H_{h}=\emptyset }[L_H, F_{H\cup K_{h-1}\cup \{h\}}(z'',z_{h},{\hat{z}})], \end{aligned}$$

\(x_{h}\in \Omega _{D,h}(y)\), \(y''=(y_1,\ldots ,y_{h-1})\), \({\hat{y}}=(y_{h+1},\ldots ,y_n)\), \(z_{h}=\alpha _{h}+i\beta _{h}\in D_{h}(z)\), \(z''=(z_1,\ldots ,z_{h-1})\), \({\hat{z}}=(z_{h+1},\ldots ,z_n)\) and \(L=(I_1,\ldots ,I_{h-1},J_{h},I_{h+1},\ldots ,I_n)\). Again from implication \((\mathrm {i})\Rightarrow (\mathrm {ii})\) of Proposition 3.13, it follows that \(\partial _{\alpha _h}F_{h,1}(z_h)=\partial _{\beta _h}F_{h,2}(z_h)\) and \(\partial _{\beta _h}F_{h,1}(z_h)=-\partial _{\alpha _h}F_{h,2}(z_h)\), i.e. \(\mathcal {D}_\epsilon f\) is slice regular w.r.t. \(\mathrm {x}_h\) on \(\Omega _{D,h}(y)\).

Conversely, assume that f is slice regular w.r.t. \(\mathrm {x}_1\) and that the functions \(\mathcal {D}^{\varepsilon (h-1)}_{\mathrm {x}_{h-1}}\cdots \mathcal {D}^{\varepsilon (1)}_{\mathrm {x}_1}f\) are slice regular w.r.t. \(\mathrm {x}_h\) for all \(K\in \mathcal {P}(n)\) and all \(h\in \{2,\ldots ,n\}\), where \(\varepsilon :\{1,\ldots ,n\}\rightarrow \{0,1\}\) is the characteristic function of K.

Let \(K\in \mathcal {P}(n)\) and let \(h\in \{1,\ldots ,n\}\) with \(h\not \in K\). Consider any \(z\in D_{K_{h-1}}\), where \(D_{K_{h-1}}=\bigcap _{k\in K_{h-1}}\{(z_1,\ldots ,z_n)\in D:z_k\not \in \mathbb {R}\}\). From formulas (27) and (28), we obtain that it holds:

$$\begin{aligned}&\textstyle \sum _{H\in \mathcal {P}(n),H_{h}=\emptyset }[L_H,\partial _{\alpha _h}F_{H\cup K_{h-1}}(z)-\partial _{\beta _h}F_{H\cup K_{h-1}\cup \{h\}}(z)]=0 \end{aligned}$$
(62)
$$\begin{aligned}&\textstyle \sum _{H\in \mathcal {P}(n),H_{h}=\emptyset }[L_H,\partial _{\beta _h}F_{H\cup K_{h-1}}(z)+\partial _{\alpha _h}F_{H\cup K_{h-1}\cup \{h\}}(z)]=0, \end{aligned}$$
(63)

where \(L=(I_1,\ldots ,I_{h-1},J_{h},I_{h+1},\ldots ,I_n)\) and \(H_0=K_0=\emptyset \).

Let \(M\in \mathcal {P}(n)\). Thanks to (4), for each \(H\in \mathcal {P}(n)\) such that \(H_h=\emptyset \), it holds

$$\begin{aligned} \partial _{\alpha _h}F_{H\cup K_{h-1}}({\overline{z}}^M)=(-1)^{|(H\cup K_{h-1})\cap M|}\partial _{\alpha _h}F_{H\cup K_{h-1}}(z); \end{aligned}$$
(64)

moreover, being \(|(H\cup K_{h-1}\cup \{h\})\cap M|=|(H\cup K_{h-1})\cap M|+|M\cap \{h\}|\), it holds

$$\begin{aligned} \partial _{\beta _h}F_{H\cup K_{h-1}{\cup \{h\}}}({\overline{z}}^M)=(-1)^{|(H\cup K_{h-1})\cap M|}\partial _{\beta _h}F_{H\cup K_{h-1}}(z). \end{aligned}$$
(65)

Thanks to (64), (65) and the validity of (62) at the point \({\overline{z}}^M\), we get that

$$\begin{aligned}&\textstyle \sum _{H\in \mathcal {P}(n),H_{h}=\emptyset }(-1)^{|(H\cup K_{h-1})\cap M|}[L_H,\partial _{\alpha _h}F_{H\cup K_{h-1}}(z)-\partial _{\beta _h}F_{H\cup K_{h-1}\cup \{h\}}(z)]=0 \end{aligned}$$
(66)

for all \(M\in \mathcal {P}(n)\). Multiply both members of (66) by \((-1)^{|K\cap M|}\) and sum over all \(M\in \mathcal {P}(n)\). Using the combinatorial Lemma 2.11, we get

$$\begin{aligned} 0&\textstyle =\sum _{M\in \mathcal {P}(n)}(-1)^{|K\cap M|} \sum _{H\in \mathcal {P}(n),H_{h}=\emptyset }(-1)^{|(H\cup K_{h-1})\cap M|}\\&\quad \textstyle \times [L_H,\partial _{\alpha _h}F_{H\cup K_{h-1}}-\partial _{\beta _h}F_{H\cup K_{h-1}\cup \{h\}}]\\&\textstyle =\sum _{H\in \mathcal {P}(n),H_{h}=\emptyset } \sum _{M\in \mathcal {P}(n)}(-1)^{|K\cap M|+|(H\cup K_{h-1})\cap M|}\\&\quad \textstyle \times [L_H,\partial _{\alpha _h}F_{H\cup K_{h-1}}-\partial _{\beta _h}F_{H\cup K_{h-1}\cup \{h\}}]\\&\quad \textstyle =\sum _{H\in \mathcal {P}(n),H_{h}=\emptyset } 2^n\delta _{K,H\cup K_{h-1}}[L_H,\partial _{\alpha _h}F_{H\cup K_{h-1}}-\partial _{\beta _h}F_{H\cup K_{h-1}\cup \{h\}}]\\&\quad \textstyle =2^n[L_{K\setminus K_h},\partial _{\alpha _h}F_{K}-\partial _{\beta _h}F_{K\cup \{h\}}] \end{aligned}$$

on the whole \(D_{K_{h-1}}\). Therefore \(\partial _{\alpha _h}F_{K}(z)=\partial _{\beta _h}F_{K\cup \{h\}}(z)\) on \(D_{K_{h-1}}\). By Assumption 3.1, we know that D is open in \(\mathbb {C}^n\). As a consequence, \(D_{K_{h-1}}\) is dense in D. Since F is of class \(\mathscr {C}^1\) on D, the partial derivatives \(\partial _{\alpha _h}F_{K}\) and \(\partial _{\beta _h}F_{K\cup \{h\}}(z)\) are continuous on D. It follows that \(\partial _{\alpha _h}F_{K}(z)=\partial _{\beta _h}F_{K\cup \{h\}}(z)\) on the whole D. In a similar way, we deduce from (63) that \(\partial _{\beta _h}F_{K}=-\partial _{\alpha _h}F_{K\cup \{h\}}\) on D. From implication \((\mathrm {ii})\Rightarrow (\mathrm {i})\) of Proposition 3.13, it follows that f is slice regular on \(\Omega _D\). \(\square \)

Leibniz’s rule

The next lemma gives sufficient conditions for Leibniz’s rule to be valid for each \(\partial _h\) and \({{\overline{\partial }}}_h\).

Lemma 3.24

Let \(\mathrm {b}=\mathcal {B}(\sigma )\) be a \(\bigtriangleup \)-product of \(\mathbb {R}^{2^n}\), let \(h\in \{1,\ldots ,n\}\) and let \(F,G:D\rightarrow A\otimes \mathbb {R}^{2^n}\) be \(\mathscr {C}^1\) stem functions. Write \(F=\sum _{K\in \mathcal {P}(n)}e_KF_K\) and \(G=\sum _{H\in \mathcal {P}(n)}e_HG_H\). Define

$$\begin{aligned} P'_{1,h}&\textstyle :=\big \{(K,H)\in \mathcal {P}(n)\times \mathcal {P}(n):\frac{\partial F_K}{\partial \beta _h}\cdot _\sigma G_H=0 \text { on }D\big \},\\ P_{1,h}&:=(\mathcal {P}(n)\times \mathcal {P}(n))\setminus P'_{1,h},\\ P'_{2,h}&\textstyle :=\big \{(K,H)\in \mathcal {P}(n)\times \mathcal {P}(n):F_K\cdot _\sigma \frac{\partial G_H}{\partial \beta _h}=0 \text { on }D\big \},\\ P_{2,h}&:=(\mathcal {P}(n)\times \mathcal {P}(n))\setminus P'_{2,h}. \end{aligned}$$

Suppose that the following two conditions hold:

$$\begin{aligned} (-1)^{|(K\bigtriangleup H)\cap \{h\}|}\sigma (K,H)=(-1)^{|K\cap \{h\}|}\sigma (K\bigtriangleup \{h\},H)\,\text { for all }(K,H)\in P_{1,h} \end{aligned}$$
(67)

and

$$\begin{aligned} (-1)^{|(K\bigtriangleup H)\cap \{h\})}\sigma (K,H)=(-1)^{|H\cap \{h\}|}\sigma (K,H\bigtriangleup \{h\})\,\text { for all }(K,H)\in P_{2,h}. \end{aligned}$$
(68)

Then it holds:

$$\begin{aligned} \partial _h(F\cdot _\sigma G)&=(\partial _hF)\cdot _\sigma G+F\cdot _\sigma (\partial _hG), \end{aligned}$$
(69)
$$\begin{aligned} {{\overline{\partial }}}_h(F\cdot _\sigma G)&=({{\overline{\partial }}}_hF)\cdot _\sigma G+F\cdot _\sigma ({{\overline{\partial }}}_hG). \end{aligned}$$
(70)

Proof

We prove only (69), the proof of (70) being similar. For simplicity, we omit ‘\(\,\cdot _\sigma \)’ in each product w.r.t \(\mathrm {b}\). We have:

$$\begin{aligned} 2 \, \partial _h(FG)=&\, \partial _{\alpha _h}(FG)-\mathcal {J}_h\big (\partial _{\beta _h}(FG)\big )=(\partial _{\alpha _h}F)G+F(\partial _{\alpha _h}G)+ \nonumber \\&\quad -\mathcal {J}_h\big ((\partial _{\beta _h}F)G\big )-\mathcal {J}_h\big (F(\partial _{\beta _h}G)\big ).\end{aligned}$$
(71)

Bearing in mind (44), we have also:

$$\begin{aligned} \mathcal {J}_h\big ((\partial _{\beta _h}F)G\big )&=\textstyle \mathcal {J}_h\big (\sum _{K,H\in \mathcal {P}(n)}e_{K\bigtriangleup H}\sigma (K,H)(\partial _{\beta _h}F_K)G_H\big )\\&=\textstyle \sum _{(K,H)\in {P_{1,h}}}\mathcal {J}_h(e_{K\bigtriangleup H})\sigma (K,H)(\partial _{\beta _h}F_K)G_H\\&=\textstyle \sum _{(K,H)\in {P_{1,h}}}e_{K\bigtriangleup H\bigtriangleup \{h\}}(-1)^{|(K\bigtriangleup H)\cap \{h\}|}\sigma (K,H)(\partial _{\beta _h}F_K)G_H. \end{aligned}$$

Similarly, we deduce:

$$\begin{aligned} \big (\mathcal {J}_h(\partial _{\beta _h}F)\big )G&=\textstyle \sum _{K,H\in \mathcal {P}(n)}\mathcal {J}_h(e_K)e_H(\partial _{\beta _h}F_K)G_H\\&=\textstyle \sum _{(K,H)\in {P_{1,h}}}e_{K\bigtriangleup \{h\}}e_H(-1)^{|K\cap \{h\}|}(\partial _{\beta _h}F_K)G_H\\&=\textstyle \sum _{(K,H)\in {P_{1,h}}}e_{K\bigtriangleup H\bigtriangleup \{h\}}(-1)^{|K\cap \{h\}|}\sigma (K\bigtriangleup \{h\},H)(\partial _{\beta _h}F_K)G_H. \end{aligned}$$

Consequently, (67) implies that \(\mathcal {J}_h\big ((\partial _{\beta _h}F)G\big )=\big (\mathcal {J}_h(\partial _{\beta _h}F)\big )G\). Similar computations and (68) ensure also that \(\mathcal {J}_h\big (F(\partial _{\beta _h}G)\big )=F\big (\mathcal {J}_h(\partial _{\beta _h}G)\big )\). Combining the latter two equalities with (71), we obtain:

$$\begin{aligned} \partial _h(FG)=&\,\textstyle \frac{1}{2}\big ((\partial _{\alpha _h}F)G+F(\partial _{\alpha _h}G) -\big (\mathcal {J}_h(\partial _{\beta _h}F)\big )G-F\big (\mathcal {J}_h(\partial _{\beta _h}G)\big )\big )\\ =&\, (\partial _hF)G+F(\partial _hG), \end{aligned}$$

as desired. \(\square \)

The above lemma suffices to prove that Leibniz’s rule works for the slice tensor product.

Proposition 3.25

For each \(f,g\in \mathcal {S}^1(\Omega _D,A)\) and for each \(h\in \{1,\ldots ,n\}\), it holds:

$$\begin{aligned} \frac{\partial }{\partial x_h}(f\odot g)&=\frac{\partial f}{\partial x_h}\odot g+f\odot \frac{\partial g}{\partial x_h}, \end{aligned}$$
(72)
$$\begin{aligned} \frac{\partial }{\partial x_h^c}(f\odot g)&=\frac{\partial f}{\partial x_h^c}\odot g+f\odot \frac{\partial g}{\partial x_h^c}. \end{aligned}$$
(73)

Proof

Given any pair \((K,H)\in \mathcal {P}(n)\times \mathcal {P}(n)\) and any \(h\in \{1,\ldots ,n\}\), it is immediate to check that \(|(K\bigtriangleup H)\cap \{h\}|+|K\cap H|=|K\cap \{h\}|+|(K\bigtriangleup \{h\})\cap H|\); indeed, the preceding equality becomes \(|K\cap H|=|K\cap H|\) if \(h\not \in K\bigtriangleup H\), and \(|K\cap H|+1=|K\cap H|+1\) if \(h\in K\bigtriangleup H\). The mentioned equality is equivalent to (67) and (68) if \(\mathrm {b}=\mathcal {B}(\sigma _\otimes ^n)\), because \(\sigma _\otimes ^n(K,H)=(-1)^{|K\cap H|}\) by Lemma 2.32. Lemma 3.24 concludes the proof. \(\square \)

Multiplication of slice regular functions

Proposition 3.26

The set \(\mathcal {SR}(\Omega _D,A)\) is a real subalgebra of \((\mathcal {S}(\Omega _D,A),\odot )\). Moreover, the set \(\mathcal {SR}_\mathbb {R}(\Omega _D,A):=\mathcal {S}_\mathbb {R}(\Omega _D,A)\cap \mathcal {SR}(\Omega _D,A)\) of all slice preserving slice regular functions from \(\Omega _D\) to A is contained in the center of \((\mathcal {S}(\Omega _D,A),\odot )\).

Proof

The first part is a direct consequence of Eq. (73) and equivalence \((\mathrm {i})\Leftrightarrow (\mathrm {i}')\) of Proposition 3.13; the second follows immediately from Lemma 2.49. \(\square \)

In the next result we see that the slice tensor product is the unique associative and hypercomplex \(\triangle \)-product on \(\mathbb {R}^{2^n}\), which preserves slice regularity.

Proposition 3.27

The tensor product \(\mathrm {b}_\otimes ^n\) is the unique associative and hypercomplex \(\bigtriangleup \)-product \(\mathrm {b}=\mathcal {B}(\sigma )\) of \(\mathbb {R}^{2^n}\) such that \(\mathcal {SR}(\Omega _D,A)\) is a real subalgebra of \((\mathcal {S}(\Omega _D,A),\cdot _\sigma )\).

Proof

Let \(\mathrm {b}=\mathcal {B}(\sigma )\) be a \(\bigtriangleup \)-product of \(\mathbb {R}^{2^n}\), let \(f=\mathcal {I}(F)\) and \(g=\mathcal {I}(G)\) be functions in \(\mathcal {SR}(\Omega _D,A)\), and let \((FG)_K\) be the K-component of \(FG=F\cdot _\sigma G\) for all \(K\in \mathcal {P}(n)\).

Choose \(K\in \mathcal {P}(n)\) and \(h\in \{1,\ldots ,n\}\) with \(h\not \in K\). For short, during the remaining part of this proof, we will use the symbol \(\sum ^*\) in place of \(\sum _{(K_1,K_2,K_3)\in \mathscr {D}(K)}\). Bearing in mind equality (40) and implication \((\mathrm {i})\Rightarrow (\mathrm {ii})\) in Proposition 3.13, we have:

$$\begin{aligned}&\partial _{\alpha _h}(FG)_K\nonumber \\&\textstyle \quad =\sum ^*\left( (\partial _{\alpha _h}F_{K_1\cup K_3})G_{K_2\cup K_3}+F_{K_1\cup K_3}(\partial _{\alpha _h}G_{K_2\cup K_3})\right) \sigma (K_1\cup K_3,K_2\cup K_3)\nonumber \\&\textstyle \quad =\sum ^*_{h\in K_3}(-\partial _{\beta _h}F_{(K_1\cup K_3)\setminus \{h\}})G_{K_2\cup K_3}\sigma (K_1\cup K_3,K_2\cup K_3)\nonumber \\&\textstyle \qquad +\sum ^*_{h\in K_3}F_{K_1\cup K_3}(-\partial _{\beta _h}G_{(K_2\cup K_3)\setminus \{h\}})\sigma (K_1\cup K_3,K_2\cup K_3)\nonumber \\&\textstyle \qquad +\sum ^*_{h\not \in K_3}(\partial _{\beta _h}F_{K_1\cup K_3\cup \{h\}})G_{K_2\cup K_3}\sigma (K_1\cup K_3,K_2\cup K_3)\nonumber \\&\textstyle \qquad +\sum ^*_{h\not \in K_3}F_{K_1\cup K_3}(\partial _{\beta _h}G_{K_2\cup K_3\cup \{h\}})\sigma (K_1\cup K_3,K_2\cup K_3)\nonumber \\&\textstyle \quad =\sum ^*_{h\not \in K_3}(-\partial _{\beta _h}F_{K_1\cup K_3})G_{K_2\cup K_3\cup \{h\}}\sigma (K_1\cup K_3\cup \{h\},K_2\cup K_3\cup \{h\}) \nonumber \\&\textstyle \qquad +\sum ^*_{h\not \in K_3}F_{K_1\cup K_3\cup \{h\}}(-\partial _{\beta _h}G_{K_2\cup K_3})\sigma (K_1\cup K_3\cup \{h\},K_2\cup K_3\cup \{h\})\nonumber \\&\textstyle \qquad +\sum ^*_{h\not \in K_3}(\partial _{\beta _h}F_{K_1\cup K_3\cup \{h\}})G_{K_2\cup K_3}\sigma (K_1\cup K_3,K_2\cup K_3)\nonumber \\&\textstyle \qquad +\sum ^*_{h\not \in K_3}F_{K_1\cup K_3}(\partial _{\beta _h}G_{K_2\cup K_3\cup \{h\}})\sigma (K_1\cup K_3,K_2\cup K_3) \end{aligned}$$
(74)

and

$$\begin{aligned}&\partial _{\beta _h}(FG)_{K\cup \{h\}}\nonumber \\&\quad \textstyle \quad =\textstyle \sum ^*_{h\not \in K_3}(\partial _{\beta _h}F_{(K_1\cup \{h\})\cup K_3})G_{K_2\cup K_3}\sigma ((K_1\cup \{h\})\cup K_3,K_2\cup K_3)\nonumber \\&\quad \textstyle \qquad +\sum ^*_{h\not \in K_3}F_{(K_1\cup \{h\})\cup K_3}(\partial _{\beta _h}G_{K_2\cup K_3})\sigma ((K_1\cup \{h\})\cup K_3,K_2\cup K_3)\nonumber \\&\quad \textstyle \qquad +\sum ^*_{h\not \in K_3}(\partial _{\beta _h}F_{K_1\cup K_3})G_{(K_2\cup \{h\})\cup K_3}\sigma (K_1\cup K_3,(K_2\cup \{h\})\cup K_3)\nonumber \\&\quad \textstyle \qquad +\sum ^*_{h\not \in K_3}F_{K_1\cup K_3}(\partial _{\beta _h}G_{(K_2\cup \{h\})\cup K_3})\sigma (K_1\cup K_3,(K_2\cup \{h\})\cup K_3)\nonumber \\&\quad \textstyle \quad =\sum ^*_{h\not \in K_3}(\partial _{\beta _h}F_{K_1\cup K_3\cup \{h\}})G_{K_2\cup K_3}\sigma (K_1\cup K_3\cup \{h\},K_2\cup K_3)+\nonumber \\&\quad \textstyle \qquad +\sum ^*_{h\not \in K_3}F_{K_1\cup K_3\cup \{h\}}(\partial _{\beta _h}G_{K_2\cup K_3})\sigma (K_1\cup K_3\cup \{h\},K_2\cup K_3)\nonumber \\&\quad \textstyle \qquad +\sum ^*_{h\not \in K_3}(\partial _{\beta _h}F_{K_1\cup K_3})G_{K_2\cup K_3\cup \{h\}}\sigma (K_1\cup K_3,K_2\cup K_3\cup \{h\})\nonumber \\&\quad \textstyle \qquad +\sum ^*_{h\not \in K_3}F_{K_1\cup K_3}(\partial _{\beta _h}G_{K_2\cup K_3\cup \{h\}})\sigma (K_1\cup K_3,K_2\cup K_3\cup \{h\}). \end{aligned}$$
(75)

It follows that, if the following chain of equalities

$$\begin{aligned} \sigma (K_1\cup K_3\cup \{h\},K_2\cup K_3\cup \{h\})&=-\sigma (K_1\cup K_3,K_2\cup K_3\cup \{h\})\nonumber \\&=-\sigma (K_1\cup K_3\cup \{h\},K_2\cup K_3)\nonumber \\&=-\sigma (K_1\cup K_3,K_2\cup K_3) \end{aligned}$$
(76)

holds for all \((K_1,K_2,K_3)\in \mathscr {D}(K)\) with \(h\not \in K_1\cup K_2\cup K_3\), then \(\partial _{\alpha _h}(FG)_K=\partial _{\beta _h}(FG)_{K\cup \{h\}}\). Similar computations ensures that, if equalities (76) hold, then \(\partial _{\beta _h}(FG)_K=-\partial _{\alpha _h}(FG)_{K\cup \{h\}}\) as well; consequently, by implication \((\mathrm {ii})\Rightarrow (\mathrm {i})\) in Proposition 3.13, \(f\cdot _\sigma g\) is slice regular. Note that if \(\sigma =\sigma _\otimes ^n\), then equalities (76) are trivially verified; indeed,

$$\begin{aligned} \sigma _\otimes ^n(K_1\cup K_3\cup \{h\},K_2\cup K_3\cup \{h\})&=(-1)^{|K_3\cup \{h\}|}=-(-1)^{|K_3|},\\ \sigma _\otimes ^n(K_1\cup K_3,K_2\cup K_3\cup \{h\})&=\sigma _\otimes ^n(K_1\cup K_3\cup \{h\},K_2\cup K_3)\\&=\sigma _\otimes ^n(K_1\cup K_3,K_2\cup K_3)=(-1)^{|K_3|}. \end{aligned}$$

This gives another proof of the fact that \(f\odot g\) is slice regular.

Suppose \(\mathrm {b}=\mathcal {B}(\sigma )\) is a hypercomplex \(\bigtriangleup \)-product of \(\mathbb {R}^{2^n}\) such that \(f\cdot _\sigma g\in \mathcal {SR}(\Omega _D,A)\) for all \(f,g\in \mathcal {SR}(\Omega _D,A)\). Let \(K,H\in \mathcal {P}(n)\). We have to prove that \(\sigma (K,H)=(-1)^{|K\cap H|}\).

First, we show that \(\sigma (K,H)=\sigma (K\cap H,K\cap H)\). Suppose that \(K\setminus H\ne \emptyset \), and choose \(h\in K\setminus H\). Denote \(x_K\) and \(x_H\) the monomial functions from \(\Omega _D\) to A defined as \(x_K:=[(x_k)_{k\in K}]\) and \(x_H:=[(x_h)_{h\in H}]\). Note that, if \(\mathcal {K}=\sum _{L\in \mathcal {P}(n)}e_L\mathcal {K}_L\) and \(\mathcal {H}=\sum _{M\in \mathcal {P}(n)}e_M\mathcal {H}_M\) denote the stem functions such that \(x_K=\mathcal {I}(\mathcal {K})\) and \(x_H=\mathcal {I}(\mathcal {H})\), then \(\mathcal {K}_L(z)=\alpha _{K\setminus L}\beta _L\) if \(L\subset K\), \(\mathcal {K}_L(z)=0\) if \(L\not \subset K\), \(\mathcal {H}_M(z)=\alpha _{H\setminus M}\beta _M\) if \(M\subset H\) and \(\mathcal {H}_M(z)=0\) if \(M\not \subset H\), where \(z=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\in D\), \(\alpha _P:=\prod _{p\in P}\alpha _p\) and \(\beta _P:=\prod _{p\in P}\beta _p\) for all \(P\in \mathcal {P}(n)\setminus \{\emptyset \}\), and \(\alpha _\emptyset =\beta _\emptyset :=1\). Moreover, by (74) and (75), if we set

$$\begin{aligned} \textstyle \sum ^\bullet&:=\textstyle \sum _{(K_1,K_2,K_3)\in \mathscr {D}((K\setminus \{h\})\bigtriangleup H),h\not \in K_3},\\ \textstyle \sum '&:=\textstyle \sum _{K_3\in \mathcal {P}(n),K_3\subset K\cap H},\\ \sigma ^{(1)}(K_3)&:=\sigma \big (((K\setminus H)\setminus \{h\})\cup K_3,(H\setminus K)\cup K_3\big ),\\ \sigma ^{(2)}(K_3)&:=\sigma \big ((K\setminus H)\cup K_3,(H\setminus K)\cup K_3\big ), \end{aligned}$$

then it holds

$$\begin{aligned}&\partial _{\alpha _h}(\mathcal {K}\mathcal {H})_{(K\setminus \{h\})\bigtriangleup H}(z)\\&\quad \textstyle \quad =\textstyle \sum ^\bullet (\partial _{\beta _h}\mathcal {K}_{K_1\cup K_3\cup \{h\}})\mathcal {H}_{K_2\cup K_3}\sigma (K_1\cup K_3,K_2\cup K_3)\\&\quad \textstyle \quad =\sum '(\alpha _{(K\cap H)\setminus K_3}\beta _{((K\setminus H)\setminus \{h\})\cup K_3})\alpha _{(K\cap H)\setminus K_3}\beta _{(H\setminus K)\cup K_3}\sigma ^{(1)}(K_3)\\&\quad \textstyle \quad =\,\beta _{(K\bigtriangleup H)\setminus \{h\}}\sum '(\alpha _{(K\cap H)\setminus K_3})^2(\beta _{K_3})^2\sigma ^{(1)}(K_3) \end{aligned}$$

and, similarly,

$$\begin{aligned} \partial _{\beta _h}(\mathcal {K}\mathcal {H})_{K\bigtriangleup H}(z)=&\textstyle \textstyle \sum ^\bullet (\partial _{\beta _h}\mathcal {K}_{K_1\cup K_3\cup \{h\}})\mathcal {H}_{K_2\cup K_3}\sigma (K_1\cup K_3\cup \{h\},K_2\cup K_3)\\ =&\textstyle \,\beta _{(K\bigtriangleup H)\setminus \{h\}}\sum '(\alpha _{(K\cap H)\setminus K_3})^2(\beta _{K_3})^2\sigma ^{(2)}(K_3). \end{aligned}$$

By hypothesis, the polynomial functions \(\partial _{\alpha _h}(\mathcal {K}\mathcal {H})_{(K\setminus \{h\})\bigtriangleup H}\) and \(\partial _{\beta _h}(\mathcal {K}\mathcal {H})_{K\bigtriangleup H}\) in the variable \(z=(\alpha _1+i\beta _1,\ldots ,\alpha _n+i\beta _n)\) are equal on the non-empty open subset D of \(\mathbb {C}^n\). Consequently, the coefficients \(\sigma ^{(1)}(K_3)\) and \(\sigma ^{(2)}(K_3)\) are equal for all \(K_3\in \mathcal {P}(n)\) with \(K_3\subset K\cap H\). In particular, the case \(K_3=K\cap H\) implies that \(\sigma (K\setminus \{h\},H)=\sigma (K,H)\) for all \(h\in K\setminus H\). Since by hypothesis \(\partial _{\beta _h}(\mathcal {K}\mathcal {H})_{(K\setminus \{h\})\bigtriangleup H}\) is equal to \(-\partial _{\alpha _h}(\mathcal {K}\mathcal {H})_{K\bigtriangleup H}\) as well, similar computations show that \(\sigma (K,H\setminus \{h\})=\sigma (K,H)\,\) for all \(h\in H\setminus K\). This proves that

$$\begin{aligned} \sigma (K,H)=\sigma (K\cap H,K\cap H)\,\text { for all }K,H\in \mathcal {P}(n), \end{aligned}$$
(77)

as desired. In particular, \(\mathrm {b}\) turns out to be commutative. Since it is also associative and hypercomplex by hypothesis, Lemma 2.42 implies that \(\mathrm {b}=\mathrm {b}_\otimes ^n\). \(\square \)

We conclude this section with a result regarding slice regularity of pointwise products. For each \(\ell \in \{1,\ldots ,n\}\), we denote \(\pi _\ell :A^n\rightarrow A\) the natural projection \(\pi _\ell (x_1,\ldots ,x_n):=x_\ell \).

Lemma 3.28

Let \(\ell \in \{1,\ldots ,n\}\), let \(E_\ell \) be an open subset of \(\mathbb {C}\) invariant under the complex conjugation of \(\mathbb {C}\) and such that \(\pi _\ell (\Omega _D)\subset E_\ell \), and let \(f:\Omega _{E_\ell }\rightarrow A\) be a slice preserving slice regular function (in one variable), that is \(f\in \mathcal {SR}_\mathbb {R}(\Omega _{E_\ell },A)\). Let \(H\in \mathcal {P}(n)\) be such that \(\ell \le h\) for all \(h\in H\), and let \(g:\Omega _D\rightarrow A\) be a H-reduced slice regular function (in n variables). Define the function \(p:\Omega _D\rightarrow A\) by

$$\begin{aligned} p(x):=f(x_\ell )g(x) \end{aligned}$$

for all \(x= (x_1,\ldots ,x_n)\in \Omega _D\), where \(f(x_\ell )g(x)\) is the product of \(f(x_\ell )\) and g(x) in A. Then p belongs to \(\mathcal {SR}(\Omega _D,A)\).

Proof

Let \(f_*:\Omega _D\rightarrow A\) be the function \(f_*(x_1,\ldots ,x_n):=f(x_h)\). Note that \(f_*\) is a \(\ell \)-reduced slice regular function. By Proposition 2.52, we have that \(p=f_*\odot g\). By (73), p is slice regular. \(\square \)

Splitting decomposition of slice regular functions

By Assumption 2.1, the set \({\mathbb {S}}_A\) is non-empty. Thanks to Artin’s theorem, the left multiplication by an element of \({\mathbb {S}}_A\) induces a complex structure on A, so the real dimension of A is even and positive, say \(\dim _\mathbb {R}(A)=2u+2\) for some \(u\in \mathbb {N}\). More precisely, if \(J\in {\mathbb {S}}_A\), then the algebraic sum of A together with the complex scalar multiplication \(\mathbb {C}_J\times A\rightarrow A\), sending (ca) into the product ca in A, defines a structure of \(\mathbb {C}_J\)-vector space. If \(\{J_1,\ldots ,J_u\}\) is a \(\mathbb {C}_J\)-vector basis of A, then \(\{1,J,J_1,JJ_1,\ldots ,J_u,JJ_u\}\) is a real vector basis of A, see [12,  Lemma 2.3]. A real vector basis of A of this form is called splitting basis of A associated with J.

In the next result we use Definition 2.44.

Proposition 3.29

Let \(f\in \mathcal {S}^1(\Omega _D,A)\), let \(J\in {\mathbb {S}}_A\) and let \(\{1,J,J_1,JJ_1,\ldots ,J_u,JJ_u\}\) be a splitting basis of A associated with J. Denote \(\{f_{k,\ell }:\Omega _D(J)\rightarrow \mathbb {R}\}_{k\in \{1,2\},\ell \in \{1,\ldots ,u\}}\) the unique real-valued functions on \(\Omega _D(J)\) such that \(f_J=\sum _{\ell =0}^u(f_{1,\ell }J_\ell +f_{2,\ell }JJ_\ell )\), where \(J_0:=1\). Define the \(\mathbb {C}_J\)-valued functions \(\{f_\ell :\Omega _D(J)\rightarrow \mathbb {C}_J\}_{\ell =0}^u\) by setting \(f_\ell :=f_{1,\ell }+f_{2,\ell }J\), in such a way that \(f_J=\sum _{\ell =0}^uf_\ell J_\ell \). The following assertions are equivalent:

  • \((\mathrm {i})\) f is slice regular.

  • \((\mathrm {ii})\) For each \(\ell \in \{0,1,\ldots ,u\}\), we have

    $$\begin{aligned} \frac{\partial f_{1,\ell }}{\partial \alpha _\ell }=\frac{\partial f_{2,\ell }}{\partial \beta _\ell }\,\,\text { and }\,\,\frac{\partial f_{1,\ell }}{\partial \beta _\ell }=-\frac{\partial f_{2,\ell }}{\partial \alpha _\ell }\,\,\text { on }\Omega _D(J), \end{aligned}$$

    where \((\alpha _1+J\beta _1,\ldots ,\alpha _n+J\beta _n)\) are the coordinates of \((\mathbb {C}_J)^n\).

  • \((\mathrm {iii})\) For each \(\ell \in \{0,1,\ldots ,u\}\), the function \(f_\ell :\Omega _D(J)\rightarrow \mathbb {C}_J\) is holomorphic, where \(\Omega _D(J)\) and \(\mathbb {C}_J\) are equipped with the natural complex structures associated with (left) multiplication by J.

Proof

Equivalence \((\mathrm {ii}) \Leftrightarrow (\mathrm {iii})\) is evident. Let \(h\in \{1,\ldots ,n\}\). Bearing in mind Artin’s theorem, we have

$$\begin{aligned} \textstyle \partial _{\alpha _h}f_J+J\partial _{\beta _h}f_J=\sum _{\ell =0}^u(\partial _{\alpha _h}f_\ell +J\partial _{\beta _h}f_\ell )J_\ell . \end{aligned}$$

Thanks to equivalence \((\mathrm {i}) \Leftrightarrow (\mathrm {iii})\) in Proposition 3.13, we deduce that f is slice regular if and only if \(\partial _{\alpha _h}f_\ell +J\partial _{\beta _h}f_\ell =0\) on \(\Omega _D\) for all \(\ell \in \{0,1,\ldots ,u\}\) and \(h\in \{1,\ldots ,n\}\). The latter assertion is in turn equivalent to say that each \(f_\ell \) is holomorphic. This proves equivalence \((\mathrm {i}) \Leftrightarrow (\mathrm {iii})\) and completes the proof. \(\square \)

As a consequence, we deduce:

Corollary 3.30

Every slice regular function is real analytic, i.e. \(\mathcal {SR}(\Omega _D,A) \subset \mathscr {C}^{\omega }(\Omega _D,A)\).

Proof

By Proposition 3.29, if \(f=\mathcal {I}(F)\in \mathcal {SR}(\Omega _D,A)\) and \(J\in {\mathbb {S}}_A\), then \(f_J\in \mathscr {C}^\omega (\Omega _D(J),A)\). By formula (8), it follows that \(F\in \mathrm {Stem}^\omega (D,A\otimes \mathbb {R}^{2^n})\) or, equivalently, \(f\in \mathcal {S}^\omega (\Omega _D,A)\). Now Theorem 2.26\((\mathrm {ii})\) ensures that \(f\in \mathscr {C}^\omega (\Omega _D,A)\). \(\square \)

Convergent power series, slice tensor and star products

In the theory of slice functions in one variable, the slice product \(f\cdot g\) between slice functions \(f=\mathcal {I}(F)\) and \(g=\mathcal {I}(G)\) is induced by the usual product between complex numbers on \(\mathbb {R}^2=\mathbb {C}\). Indeed, the latter product determines a real algebra structure on \(A\otimes \mathbb {R}^2\), which coincides with the tensor product \(A\otimes \mathbb {C}\). Then such a tensor product is used to compute the pointwise product FG and, finally, one defines \(f\cdot g:=\mathcal {I}(FG)\). One of the main features of the slice product \(f\cdot g\) is revealed by its algebraic nature: if f and g are polynomials or, more generally, convergent power series in one variable, then \(f\cdot g\) coincides with the standard abstract product \(f*g\), sometimes called star product of f and g in the noncommutative setting, see [9,  §5].

The real algebra \(A\otimes \mathbb {C}^{\otimes n}\) defines the slice tensor product ‘\(\,\odot \,\)’ on slice functions in n variables, see Definition 2.36. In this way, it is natural to ask whether such a product coincides with the star product on polynomials and on convergent power series in n-variables as well.

The aim of this section is to answer affirmatively to this question.

Let us review the concept of convergent power series with coefficients in A. A formal power series s in n indeterminates \(X=(X_1,\ldots ,X_n)\) with coefficients in A is a sequence \(\{a_\ell \}_{\ell \in \mathbb {N}^n}\) of elements of A. We formally write: \(s(X)=\sum _{\ell \in \mathbb {N}^n}X^\ell a_\ell \) and \(X^\ell =X_1^{\ell _1}\cdots X_n^{\ell _n}\) if \(\ell =(\ell _1,\ldots ,\ell _n)\). If the set \(\{\ell \in \mathbb {N}^n:a_\ell \ne 0\}\) is finite, then s is called a (formal) polynomial. Denote \(A[[X]]=A[[X_1,\ldots ,X_n]]\) the set of all such formal power series. We can add and multiply formal power series in a standard way: if \(t(X)=\sum _{\ell \in \mathbb {N}^n}X^\ell b_\ell \in A[[X]]\), then \(s+t\) and \(s*t\) are the elements of A[[X]] defined by

$$\begin{aligned}&(s+t)(X)\textstyle :=\sum _{\ell \in \mathbb {N}^n}X^\ell (a_\ell +b_\ell ),\\&\quad (s*t)(X)\textstyle :=\sum _{\ell \in \mathbb {N}^n}X^\ell \big (\sum _{p,q\in \mathbb {N}^n,\,p+q=\ell }a_pb_q\big ). \end{aligned}$$

Given \(r\in \mathbb {R}\), we can also define the real scalar multiplication sr by \((sr)(X):=\sum _{\ell \in \mathbb {N}^n}X^\ell (a_\ell r)\). These operations make A[[X]] a real algebra.

Assumption 3.31

In what follows, we assume that \(\Vert \cdot \Vert :A\rightarrow \mathbb {R}\) is a norm of A such that

$$\begin{aligned} \Vert x\Vert =\sqrt{n(x)}\,\,\text { for all }x\in Q_A, \end{aligned}$$
(78)

equivalently, \(\Vert \alpha +J\beta \Vert =\sqrt{\alpha ^2+\beta ^2}\) for all \(\alpha ,\beta \in \mathbb {R}\) and \(J\in {\mathbb {S}}_A\).

Examples of real alternative \(^*\)-algebra A with a norm having property (78) are as follows: \(\mathbb {H}\) and \(\mathbb {O}\) with the usual conjugation \(x^c:={\overline{x}}\) and the usual Euclidean norm; the real Clifford algebra \(\mathbb {R}_n\) with signature (0, n), with the Clifford conjugation [17,  Definition 3.7], and the usual Euclidean norm of \(\mathbb {R}^{2^n}\) or the Clifford operator norm, as defined in [16,  (7.20)].

Assumption 3.31 implies that

$$\begin{aligned} {\mathbb {S}}_A\text { is compact in }A. \end{aligned}$$
(79)

Indeed, \({\mathbb {S}}_A=\{I\in A:t(I)=0,n(I)=1\}\) is closed in A and it is contained in the compact subset \(S:=\{a\in A:\Vert a\Vert =1\}\) of A.

We say that \(s(X)=\sum _{\ell \in \mathbb {N}^n}X^\ell a_\ell \in A[[X]]\) is a convergent power series if there exists a real number \(M>0\) such that

$$\begin{aligned} \Vert a_\ell \Vert \le M^{|\ell |} \,\, \text { for all } \ell =(\ell _1,\ldots ,\ell _n)\in \mathbb {N}^n\text { with } \textstyle |\ell |=\sum _{h=1}^n\ell _h. \end{aligned}$$
(80)

Note that such a concept does not depend on the chosen norm of A. Indeed, A has finite real dimension so all the norms of A are equivalent. Define the real number \(B:=\max _{x,y\in S}\Vert xy\Vert \). Note that \(B>0\); indeed, \(B\ge \Vert (1\Vert 1\Vert ^{-1})\cdot (1\Vert 1\Vert ^{-1})\Vert =\Vert 1\Vert ^{-1}>0\). By the very definition of B, it follows at once that

$$\begin{aligned} \Vert xy\Vert \le B\Vert x\Vert \Vert y\Vert \,\text { for all }x,y\in A. \end{aligned}$$
(81)

Thanks to the latter inequality, it follows immediately that the set of all convergent power series is a real subalgebra of A[[X]].

Suppose now that \(s(X)=\sum _{\ell \in \mathbb {N}^n}X^\ell a_\ell \) satisfies (80). Let \(\rho \in \mathbb {R}\) with \(0<B\rho M=:\gamma <1\) and let \(\mathbb {B}_\rho :=\bigcap _{h=1}^n\{(x_1,\ldots ,x_n)\in A^n:\Vert x_h\Vert <\rho \}\). Note that, if \(x\in \mathbb {B}_\rho \) and \(x^\ell a_\ell \in A\) is defined as in Definition 2.7, then \(\Vert x^\ell a_\ell \Vert \le B^{|\ell |}\rho ^{|\ell |}\Vert a_\ell \Vert \le \gamma ^{|\ell |}\); consequently, the series \(\sum _{\ell \in \mathbb {N}^n}\Vert x^\ell a_\ell \Vert =\sum _{h\in \mathbb {N}}(\sum _{\ell \in \mathbb {N}^n,|\ell |=h}\Vert x^\ell a_\ell \Vert )\) is dominated by the positive real term series \(\sum _{h\in \mathbb {N}}(h+1)^n\gamma ^h\), which converges in \(\mathbb {R}\). This proves that, for each \(x\in \mathbb {B}_\rho \), the series \(\sum _{\ell \in \mathbb {N}^n}x^\ell a_\ell \) converges in A to a point s(x). We abuse notation denoting \(s:\mathbb {B}_\rho \rightarrow A\) the function from \(\mathbb {B}_\rho \) to A, sending x into s(x). We say that such a function s is a sum of the convergent power series s(X). Note that the series \(\sum _{\ell \in \mathbb {N}^n}x^\ell a_\ell \) converges to \(s:\mathbb {B}_\rho \rightarrow A\) uniformly on \(\mathbb {B}_\rho \).

Proposition 3.32

Let \(s:\mathbb {B}_\rho \rightarrow A\) be a sum of a convergent power series s(X). Then s is a slice regular function and there exists a unique sequence \(\{a_\ell \}_{\ell \in \mathbb {N}^n}\) in A such that \(s(x)=\sum _{\ell \in \mathbb {N}^n}x^\ell a_\ell \). In particular, this is true if s is a polynomial function.

Proof

Corollary 2.17 ensures that s is a slice function. By Proposition 3.14, the monomial function \(M_\ell :\mathbb {B}_\rho \rightarrow A\), sending x into \(x^\ell a_\ell \), is slice regular for each \(\ell \in \mathbb {N}^n\). Choose \(J\in {\mathbb {S}}_A\). By Proposition 3.13, we know that \(\partial _{\alpha _h}M_{\ell ,J}+J\partial _{\beta _h}M_{\ell ,J}=0\) on \(\mathrm {B}:=\mathbb {B}_\rho \cap (\mathbb {C}_J)^n\), where \(M_{\ell ,J}\) is the restriction of \(M_\ell \) to \(\mathrm {B}\). Since the series \(\sum _{\ell \in \mathbb {N}^n}M_{\ell ,J}\) converges to \(s_J\) uniformly on \(\mathrm {B}\) and both the series \(\sum _{\ell \in \mathbb {N}^n}\partial _{\alpha _h}M_{\ell ,J}\) and \(\sum _{\ell \in \mathbb {N}^n}\partial _{\beta _h}M_{\ell ,J}\) converge uniformly on \(\mathrm {B}\) as well, for all \(h\in \{1,\ldots ,n\}\), we can differentiate \(s_J\) term by term obtaining:

$$\begin{aligned} \textstyle \partial _{\alpha _h}s_J+J\partial _{\beta _h}s_J=\sum _{\ell \in \mathbb {N}^n}(\partial _{\alpha _h}M_{\ell ,J}+J\partial _{\beta _h}M_{\ell ,J})=0 \end{aligned}$$

on \(\mathrm {B}\). Using Proposition 3.13 again, we deduce that s is slice regular. Finally, note that \(a_\ell =(\ell !)^{-1}D_\ell s(0)\), where \(D_\ell \) denotes the partial derivative \(\partial ^{|\ell |}/\partial \alpha _1^{\ell _1}\cdots \partial \alpha _n^{\ell _n}\). \(\square \)

Remark 3.33

In the polynomial case, the uniqueness assertion in the statement of the preceding result can be improved. Let \(s:\Omega _D\rightarrow A\) be a polynomial function. If D is open in \(\mathbb {C}^n\), then there exists a unique sequence \(\{a_\ell \}_{\ell \in \mathbb {N}^n}\) in A such that the set \(L:=\{\ell \in \mathbb {N}^n:a_\ell \ne 0\}\) is finite and \(s(x)=\sum _{\ell \in L}x^\ell a_\ell \) on \(\Omega _D\). As s is a polynomial, \(s(x)=\sum _{\ell \in L}x^\ell a_\ell \) for some finite non-empty subset L of \(\mathbb {N}^n\) and for some \(a_\ell \) in A. Assume \(s=0\) on \(\Omega _D\). We have to show that each \(a_\ell \) is equal to zero. By Assumption 2.1, we know that D and \({\mathbb {S}}_A\) are non-empty. Let \(J\in {\mathbb {S}}_A\). Let \(z=(z_1,\ldots ,z_n)\in \Omega _D(J)\), let \(\ell =(\ell _1,\ldots ,\ell _n)\in \mathbb {N}^n\) and \(m=(m_1,\ldots ,m_n)\in \mathbb {N}^n\). Bearing in mind Artin’s theorem and the fact that the components \(z_h\) of z commute each other, we have that \(z^\ell a_\ell =[(z_h^{\ell _h})_{h=1}^n,a_\ell ]=(z^\ell )a_\ell \) and the complex derivative \(D_ms_J:\Omega _D(J)\rightarrow A\) of \(s_J\) is well-defined and it is equal to \(\sum _{\ell \in \mathbb {N}^n}\frac{\ell !}{(\ell -m)!}(z^{\ell -m})a_\ell \), where \(D_ms_J:=\partial ^{|m|}s_J/\partial z_1^{m_1}\cdots \partial z_n^{m_n}\). Since \(s_J=0\) on \(\Omega _D(J)\), the same is true for each complex derivative \(D_ms_J\), i.e. \(D_ms_J=0\) on \(\Omega _D(J)\). We prove by induction on the cardinality of L that \(a_\ell =0\) for all \(\ell \in L\). If L is the singleton \(\{\ell \}\), then \(a_\ell =D_\ell s_J=0\). Suppose L contains at least two elements. Choose \(m\in L\) such that \(|m|\ge |\ell |\) for all \(\ell \in L\). Since \(\frac{D_ms_J}{m!}=a_m\), we have that \(a_m=0\) and \(s_J(z)=\sum _{\ell \in L\setminus \{m\}}(z^\ell )a_\ell \). The cardinality of \(L\setminus \{m\}\) is \(l-1\) so by induction all \(a_\ell =0\). \(\square \)

Definition 3.34

Let \(s:\mathbb {B}_{\rho _1}\rightarrow A\) and \(t:\mathbb {B}_{\rho _2}\rightarrow A\) be sums of convergent power series s(X) and t(X), respectively. Let \(\{a_p\}_{p\in \mathbb {N}^n}\) and \(\{b_q\}_{q\in \mathbb {N}^n}\) be the unique sequences in A such that \(s(X)=\sum _{p\in \mathbb {N}^n}X^p a_p\) and \(t(X)=\sum _{q\in \mathbb {N}^n}X^q b_q\), see Proposition 3.32. Given a sum \(s*t:\mathbb {B}_{\rho _3}\rightarrow A\) of the convergent power series \((s*t)(X)\), we say that \(s*t\) is a star product of s and t.

Suppose that D is open in \(\mathbb {C}^n\), and \(s,t:\Omega _D\rightarrow A\) are polynomial functions. Let \(\{a_p\}_{p\in \mathbb {N}^n}\) and \(\{b_q\}_{q\in \mathbb {N}^n}\) be the unique sequences in A such that \(s(X)=\sum _{p\in \mathbb {N}^n}X^p a_p\) and \(t(X)=\sum _{q\in \mathbb {N}^n}X^q b_q\), see Remark 3.33. The star product \(s*t:\Omega _D\rightarrow A\) of s and t is defined as for s(X) and t(X), that is \((s*t)(x):=\sum _{\ell \in \mathbb {N}^n}x^\ell (\sum _{p,q\in \mathbb {N}^n,\,p+q=\ell }a_pb_q)\). \(\square \)

Proposition 3.35

The following hold.

  • \((\mathrm {i})\) Let \(s,t:\mathbb {B}_\rho \rightarrow A\) be sums of convergent power series and let \(s*t:\mathbb {B}_{\rho '}\rightarrow A\) be a star product of s and t with \(0<\rho '\le \rho \). Then \(s\odot t=s*t\) on \(\mathbb {B}_{\rho '}\).

  • \((\mathrm {ii})\) Let \(s,t:\Omega _D\rightarrow A\) be polynomial functions. Suppose D is open in \(\mathbb {C}^n\). Then \(s\odot t=s*t\) on \(\Omega _D\). In particular, given any \(\ell ,m\in \mathbb {N}^n\) and \(a,b\in A\), we have:

    $$\begin{aligned} (x^\ell a)\odot (x^mb)=(x^\ell a)*(x^mb)=x^{\ell +m}(ab)\,\, \text { on }\Omega _D. \end{aligned}$$
    (82)

Proof

Let us start proving \((\mathrm {ii})\). Equip \(\mathcal {S}(\Omega _D,A)\) with the slice tensor product ‘\(\,\odot \,\)’. Write ‘\(\,\cdot \,\)’ in place ‘\(\,\odot \,\)’ for short. Let \(\ell =(\ell _1,\ldots ,\ell _n)\) and \(m=(m_1,\ldots ,m_n)\). Since the slice tensor product is hypercomplex, we can apply Lemma 2.43. As a consequence, in order to complete the proof, it suffices to show that \((x^{\bullet \ell }\cdot a)\cdot (x^{\bullet m}\cdot b)=x^{\bullet \ell +m}\cdot (ab)\). Let \(h\in \{1,\ldots ,n\}\). Note that the stem function \(F=F_\emptyset +e_hF_h\) inducing \(x_h^{\bullet \ell _h}\) has real-valued components \(F_\emptyset \) and \(F_h\). The same is true for \(x_h^{\bullet m_h}\). By Lemma 2.49, it follows at once that \(x_h^{\bullet \ell _h}\) and \(x_h^{\bullet m_h}\) belong to the center of \(\mathcal {S}(\Omega _D,A)\). Consequently, it holds \(x^{\bullet \ell }\cdot x^{\bullet m}=x^{\bullet \ell +m}\) and

$$\begin{aligned} (x^{\bullet \ell }\cdot a)\cdot (x^{\bullet m} \cdot b)=(x^{\bullet \ell }\cdot x^{\bullet m})\cdot (ab)=x^{\bullet \ell +m}\cdot (ab). \end{aligned}$$

This proves (82) and hence point \((\mathrm {ii})\). Passing \((\mathrm {ii})\) to the limit, we obtain \((\mathrm {i})\). \(\square \)

Slice regular functions and ordered analyticity

Let \(\Vert \cdot \Vert :A\rightarrow \mathbb {R}\) be a norm of A satisfying Assumption 3.31, let \(\rho \in \mathbb {R}\) with \(\rho >0\) and let \(\mathbb {B}_\rho :=\bigcap _{h=1}^n\{(x_1,\ldots ,x_n)\in A^n:\Vert x_h\Vert <\rho \}\), as in Sect. 3.8.

Theorem 3.36

A function \(f:\mathbb {B}_\rho \rightarrow A\) is slice regular if and only if f is a sum of a convergent power series \(\sum _{\ell \in \mathbb {N}^n}X^\ell a_\ell \) with coefficients in A. Moreover, if f is slice regular, then

$$\begin{aligned} a_\ell =(\ell !)^{-1}\partial _\ell f(0) \end{aligned}$$

for all \(\ell =(\ell _1,\ldots ,\ell _n)\in \mathbb {N}^n\), where \(\partial _\ell \) is the derivative \(\partial ^{|\ell |}/\partial x_1^{\ell _1}\cdots \partial x_n^{\ell _n}\) obtained by composing in any order \(\ell _1\)-times \(\partial /\partial x_1\), \(\ell _2\)-times \(\partial /\partial x_2\), \(\ldots \), \(\ell _n\)-times \(\partial /\partial x_n\).

Proof

By Proposition 3.32, if \(f(x)=\sum _{\ell \in \mathbb {N}^n}x^\ell a_\ell \) on \(\mathbb {B}_\rho \), then f is slice regular. Suppose that f is slice regular. Let \(J\in {\mathbb {S}}_A\), let \(\{1,J,J_1,JJ_1,\ldots ,J_u,JJ_u\}\) be a splitting basis of A associated with J and let \(\{f_h:\mathbb {B}_\rho (J)\rightarrow \mathbb {C}_J\}_{h=1}^u\) be the family of \(\mathbb {C}_J\)-complex functions such that \(f_J=\sum _{h=0}^uf_hJ_h\), where \(J_0:=1\). By Proposition 3.29, we know that each \(f_h\) is holomorphic. As a consequence, each \(f_h\) is the sum of a (unique) series \(\sum _{\ell \in \mathbb {N}^n}x^\ell a_{\ell ,h}\) with coefficients in \(\mathbb {C}_J\), converging totally on compact subsets of \(\mathbb {B}_\rho (J)\) in the sense that \(\sum _{\ell \in \mathbb {N}^n}\max _{x\in C}\Vert x^\ell a_{\ell ,h}\Vert <+\infty \) for all compact subsets C of \(\mathbb {B}_\rho (J)\). Set \(a_\ell :=\sum _{h=0}^ua_{\ell ,h}J_h\). Bearing in mind Artin’s theorem, we deduce that \(f_J\) is the sum of the series \(\sum _{\ell _\in \mathbb {N}^n}x^\ell a_\ell \), converging totally on compact subsets of \(\mathbb {B}_\rho (J)\). In particular, if \(M_\ell :\mathbb {B}_\rho \rightarrow A\) denotes the monomial function \(x^\ell a_\ell \) for each \(\ell \in \mathbb {N}^n\), then \(Q_C:=\sum _{\ell \in \mathbb {N}^n}\max _{x\in C}\Vert M_\ell (x)\Vert \) is finite for all compact subsets C of \(\mathbb {B}_\rho (J)\). Choose arbitrarily a non-empty circular compact subset \(C^*\) of \(\mathbb {B}_\rho \) and let \(C:=C^*\cap (\mathbb {C}_J)^n\subset \mathbb {B}_\rho (J)\). Let \(x=(\alpha _1+J\beta _1,\ldots ,\alpha _n+J\beta _n)\in C\) and let \(y=(\alpha _1+I_1\beta _1,\ldots ,\alpha _n+I_n\beta _n)\in C^*\) for some \(I=(I_1,\ldots ,I_n)\in ({\mathbb {S}}_A)^n\). By (79), we know that \({\mathbb {S}}_A\) is compact in A so \(L:=\max \{\Vert 1\Vert ,\max _{a\in {\mathbb {S}}_A}\Vert a\Vert \}\) is a positive real number. Let B be a positive real number with property (81). Let \(\ell \in \mathbb {N}^n\). By representation formula (7), we have that \(M_\ell (y)=2^{-n}\sum _{K,H \in \mathcal {P}(n)}(-1)^{|K \cap H|}[I_K,[J^{-|K|}, M_\ell (x^{\, c,H})]]\). Consequently,

$$\begin{aligned} \Vert M_\ell (y)\Vert&\le \textstyle 2^{-n}\sum _{K,H \in \mathcal {P}(n)}(BL)^{|K|+1}\Vert M_\ell (x^{\, c,H})\Vert \\&\le \textstyle 2^{-n}(BL)^{n+1}\sum _{K\in \mathcal {P}(n)}\big (\sum _{H \in \mathcal {P}(n)}\Vert M_\ell (x^{\, c,H})\Vert \big )\\&\le \textstyle (BL)^{n+1}\sum _{H \in \mathcal {P}(n)}\Vert M_\ell (x^{\, c,H})\Vert \end{aligned}$$

and hence

$$\begin{aligned} \textstyle \max _{y\in C^*}\Vert M_\ell (y)\Vert&\le \textstyle (BL)^{n+1}\sum _{H \in \mathcal {P}(n)}\max _{x\in C}\Vert M_\ell (x^{\, c,H})\Vert \\&\textstyle =(BL)^{n+1}\sum _{H \in \mathcal {P}(n)}\max _{x\in C}\Vert M_\ell (x)\Vert \\&\textstyle =2^n(BL)^{n+1}\max _{x\in C}\Vert M_\ell (x)\Vert . \end{aligned}$$

As a consequence, we deduce:

$$\begin{aligned} \textstyle \sum _{\ell \in \mathbb {N}^n}\max _{y\in C^*}\Vert M_\ell (y)\Vert&\le \textstyle 2^n(BL)^{n+1}Q_{C}<+\infty . \end{aligned}$$

This proves that the series \(\sum _{\ell \in \mathbb {N}^n}x^\ell a_\ell \) converges to a function \(s:\mathbb {B}_\rho \rightarrow A\), totally on compact subsets of \(\mathbb {B}_\rho \). Since f and s are slice functions which coincide on \(\mathbb {B}_\rho (J)\), f and s coincide on the whole \(\mathbb {B}_\rho \) by Corollary 2.13. Finally, differentiating s term by term, we obtain that \(\partial _\ell f(0)=\partial _\ell s(0)=\ell !a_\ell \). We are done. \(\square \)

Cauchy integral formula for slice regular functions

Throughout this section, we suppose Assumption 3.1 is true, i.e. D is open in \(\mathbb {C}^n\). Moreover, we fix \(J\in {\mathbb {S}}_A\).

Some preparations

Recall that \(\phi _J:\mathbb {C}\rightarrow \mathbb {C}_J\) is the real algebra isomorphism \(\phi _J(\alpha +i\beta ):=\alpha +J\beta \). Choose bounded open subsets \(E'_1,\ldots ,E'_n\) of \(\mathbb {C}\) invariant under complex conjugation and with \(\mathscr {C}^1\) boundaries \(\partial E'_1,\ldots ,\partial E'_n\). Let \(h\in \{1,\ldots ,n\}\). Define \(E_h:=\phi _J(E_h')\) and \(\partial E_h:=\phi _J(\partial E_h')\). Note that \(\partial E_h\) is the boundary of \(E_h\) in \(\mathbb {C}_J\). Moreover, \(\partial E_h\) is the disjoint union of a certain finite number, say \(c_h\), of connected components, each homeomorphic to the circumference \({\mathbb {S}}^1\). Choose a \(\mathscr {C}^1\) parametrization \(\xi _h:T_h\rightarrow \partial E_h\) of \(\partial E_h\). Here \(T_h\) is the disjoint union of \(c_h\) intervals of the form \(\{[a_{h,l},b_{h,l}]\}_{l=1}^{c_h}\), \(\{\xi _h([a_{h,l},b_{h,l}])\}_{l=1}^{c_h}\) is the family of all connected components of \(\partial E_h\), \(\xi _h(a_{h,l})=\xi _h(b_{h,l})\) for all \(l\in \{1,\ldots ,c_h\}\), and each restriction of \(\xi _h\) to \([a_{h,l},b_{h,l})\) is injective.

Consider the open subset E of \((\mathbb {C}_J)^n\) and its distinguished boundary \(\partial ^*E\) given by

$$\begin{aligned} E:=E_1\times \cdots \times E_n\,\,\,\text { and }\,\,\,\partial ^*E:=(\partial E_1)\times \cdots \times (\partial E_n). \end{aligned}$$

Define

$$\begin{aligned} \Omega (E):=\Omega _{E'_1\times \ldots \times E'_n}. \end{aligned}$$

Note that \(E=\Omega (E)\cap (\mathbb {C}_J)^n\), i.e. \(\Omega (E)\) is the smallest circular set containing E. Denote \(\mathrm {cl}(\Omega (E))\) the closure of \(\Omega (E)\) in \((Q_A)^n\).

Given \(v=(v_1,\ldots ,v_n)\in (\mathbb {C}_J)^n\) and \(K\in \mathcal {P}(n)\), we define the element \(v_K^c\) of A by setting

$$\begin{aligned} v_K^c:= \left\{ \begin{array}{ll} 1 &{} \text { if }K=\emptyset , \\ \textstyle \prod _{h\in K}(v_h)^c &{} \text { if }K\ne \emptyset . \end{array} \right. \end{aligned}$$

Let \(T:=T_1\times \cdots \times T_n\). We define the maps \(\xi :T\rightarrow \partial ^*E\) and \({\dot{\xi }}:T\rightarrow \mathbb {C}_J\) by

$$\begin{aligned} \xi (t):=(\xi _1(t_1),\ldots ,\xi _n(t_n)) \,\,\,\text { and }\,\,\, {\dot{\xi }}(t):={\dot{\xi }}_1(t_1)\cdots {\dot{\xi }}_n(t_n) \end{aligned}$$

for all \(t=(t_1,\ldots ,t_n)\in T\), where \({\dot{\xi }}_h(t_h)\in \mathbb {C}_J\) denotes the derivative of \(\xi _h\) at \(t_h\) in \(\mathbb {C}_J\), and \({\dot{\xi }}_1(t_1)\cdots {\dot{\xi }}_n(t_n)\) the product of the \({\dot{\xi }}_h(t_h)\)’s in \(\mathbb {C}_J\) or, equivalently, in A. Given any \(K\in \mathcal {P}(n)\), we define also the function \(\xi _K^c:T\rightarrow \mathbb {C}_J\) by \(\xi _K^c(t):=(\xi (t))_K^c\), i.e.,

$$\begin{aligned} \xi _K^c(t):= \left\{ \begin{array}{ll} 1 &{} \text { if }K=\emptyset , \\ \textstyle \prod _{h\in K}(\xi _h(t_h))^c &{} \text { if }K\ne \emptyset , \end{array} \right. \end{aligned}$$

for all \(t=(t_1,\ldots ,t_n)\in T\).

We need also two variants of Definition 2.4. Let \(C:\{1,\ldots ,n\}\rightarrow A\) be any function and let \(a\in A\). We define the elements [Ca] and [C] of A by setting

$$\begin{aligned} \textstyle [C,a]:=C(1)\big (C(2)\big (C(3)\cdots \big (C(n-1)\big (C(n)a\big )\big )\ldots \big )\big ) \end{aligned}$$

and

$$\begin{aligned} \textstyle [C]:=[C,1]=C(1)\big (C(2)\big (C(3)\cdots \big (C(n-1)C(n)\big )\ldots \big )\big ). \end{aligned}$$

In addition, given any map \(\mathcal {C}:\{1,\ldots ,n\}\rightarrow \mathcal {S}(\Omega (E),A)\), we define the slice functions \([\mathcal {C},a]\) and \(\mathcal {C}\) in \(\mathcal {S}(\Omega (E),A)\) by setting

$$\begin{aligned} \textstyle [\mathcal {C},a]_\odot :=\mathcal {C}(1)\odot \big (\mathcal {C}(2)\odot \big (\mathcal {C}(3)\cdots \odot \big (\mathcal {C}(n-1)\odot \big (\mathcal {C}(n)\odot a\big )\big )\ldots \big )\big ) \end{aligned}$$

and

$$\begin{aligned} \textstyle {[}\mathcal {C}]_\odot :=[\mathcal {C},1]_\odot =\mathcal {C}(1)\odot \big (\mathcal {C}(2)\odot \big (\mathcal {C}(3)\cdots \odot \big (\mathcal {C}(n-1)\odot \mathcal {C}(n)\big )\ldots \big )\big ); \end{aligned}$$

here a (in particular 1) is identified with the function \(\Omega (E)\rightarrow A\) constantly equal to a.

Recall that, given any \(q\in Q_A\), the function \(\Delta _q:Q_A\rightarrow A\) is the characteristic polynomial of q, i.e. \(\Delta _q(p):=p^2-2\mathrm {Re}(q)p+n(q)\). Denote \(\Gamma _n\) the non-empty open subset of \((Q_A)^n\times (Q_A)^n\) defined by

$$\begin{aligned} \textstyle \Gamma _n:=\bigcap _{h=1}^n\{((x_1,\ldots ,x_n),(y_1,\ldots ,y_n))\in (Q_A)^n\times (Q_A)^n:\Delta _{y_h}(x_h)\ne 0\}, \end{aligned}$$
(83)

and \(\Gamma _n(J)\) the non-empty open subset of \((Q_A)^n\times (\mathbb {C}_J)^n\) given by

$$\begin{aligned}&\Gamma _n(J):=\Gamma _n\cap ((Q_A)^n\times (\mathbb {C}_J)^n). \end{aligned}$$
(84)

The general case

Let \(y=(y_1,\ldots ,y_n)\in \partial ^*E\). For each \(h\in \{1,\ldots ,n\}\), let \(\mathcal {C}_{y,h}:\Omega (E)\rightarrow A\) be the Cauchy kernel in the variable \(x_h\) w.r.t. \(y_h\), i.e. the slice function defined as follows:

$$\begin{aligned} \mathcal {C}_{y,h}(x):=\Delta _{y_h}(x_h)^{-1}(y_h^c-x_h)\,\,\text { for all }x=(x_1,\ldots ,x_n)\in \Omega (E), \end{aligned}$$
(85)

where \(y_h^c:=(y_h)^c\) and \(\Delta _{y_h}(x_h)^{-1}\) is the inverse of \(\Delta _{y_h}(x_h)\) in \(\mathbb {C}_J\) or, equivalently, in A. We define \(\mathcal {C}_y:\{1,\ldots ,n\}\rightarrow \mathcal {S}(\Omega (E),A)\) by

$$\begin{aligned} \mathcal {C}_y(h):=\mathcal {C}_{y,h}\,\,\text { for each } h\in \{1,\ldots ,n\}. \end{aligned}$$
(86)

Definition 3.37

Let S be any subset of \(A^n\) containing \(\partial ^*E\) and let \(g:S\rightarrow A\) be any function. We define the function \(C_g:\Omega (E)\times T\rightarrow A\) by setting

$$\begin{aligned} C_g(x,t):=[\mathcal {C}_{\xi (t)},{{\dot{\xi }}(t)}J^{-n}g(\xi (t))]_\odot (x). \, \end{aligned}$$
(87)

\(\square \)

Remark 3.38

For each \(t\in T\), the function \(\Omega (E)\rightarrow A\), \(x\mapsto C_g(x,t)\) is slice regular. This follows immediately from Proposition 3.26 and the fact that each function \(\mathcal {C}_{\xi (t),h}:\Omega (E)\rightarrow A\) and the constant function \(\Omega (E)\rightarrow A\), \(x\mapsto {\dot{\xi }}(t)J^{-n}g(\xi (t))\) are slice regular. \(\square \)

Our general Cauchy integral formula reads as follows.

Theorem 3.39

Let \(f:\Omega _D\rightarrow A\) be a slice regular function. Suppose that \(\mathrm {cl}(\Omega (E))\subset \Omega _D\). Then

$$\begin{aligned} f(x)=(2\pi )^{-n}\int _TC_f(x,t)\,dt\quad \text { for all }x\in \Omega (E), \end{aligned}$$
(88)

where \(dt=dt_1\cdots dt_n\).

Proof

Let \(\{1,J,J_1,JJ_1,\ldots ,J_u,JJ_u\}\) be a splitting basis of A, see above Sect. 3.7. Write \(f_J:E\rightarrow A\) as follows: \(f_J=\sum _{\ell =0}^uf_\ell J_\ell \) for some (unique) functions \(f_\ell :E\rightarrow \mathbb {C}_J\). By Proposition 3.29, we know that each \(f_\ell \) is holomorphic. In this way, given any \(x=(x_1,\ldots ,x_n)\in E\), we can apply the classical Cauchy formula to \(f_\ell \) obtaining

$$\begin{aligned} f_\ell (x)&=(2\pi J)^{-n}\int _{\partial ^*E}\frac{f_\ell (y)}{(y_1-x_1)\cdots (y_n-x_n)}\,dy_1\cdots dy_n\\&=(2\pi )^{-n}\int _{\partial ^*E}\mathcal {C}_{y,1}(x)\cdots \mathcal {C}_{y,n}(x)J^{-n}f_\ell (y)\,dy_1\cdots dy_n\\&=(2\pi )^{-n}\int _T\mathcal {C}_{\xi (t),1}(x)\cdots \mathcal {C}_{\xi (t),n}(x)J^{-n}f_\ell (\xi (t)){\dot{\xi }}(t)\,dt\\&=(2\pi )^{-n}\int _T\mathcal {C}_{\xi (t),1}(x)\cdots \mathcal {C}_{\xi (t),n}(x){\dot{\xi }}(t)J^{-n}f_\ell (\xi (t))\,dt. \end{aligned}$$

Thanks to Artin’s theorem, we deduce

$$\begin{aligned} f_J(x)&=\sum _{\ell =0}^u\left( (2\pi )^{-n}\int _T\mathcal {C}_{\xi (t),1}(x)\cdots \mathcal {C}_{\xi (t),n}(x){\dot{\xi }}(t)J^{-n}f_\ell (\xi (t))\,dt\right) J_\ell \\&=(2\pi )^{-n}\int _T\mathcal {C}_{\xi (t),1}(x)\cdots \mathcal {C}_{\xi (t),n}(x){\dot{\xi }}(t)J^{-n}f(\xi (t))\,dt. \end{aligned}$$

Choose \(t\in T\) and define \(\psi _t\in \mathcal {S}(\Omega (E),A)\) by \(\psi _t(x):=C_f(x,t)\). Bearing in mind Lemma 2.47, Proposition 2.48 and Definition 3.37, we have:

$$\begin{aligned} (\psi _t)_J&=\big ([\mathcal {C}_{\xi (t)},{\dot{\xi }}(t)J^{-n}f(\xi (t))]_\odot \big )_J\\&=\big ((\mathcal {C}_{\xi (t),1}\odot \cdots \odot \mathcal {C}_{\xi (t),n})\odot ({\dot{\xi }}(t)J^{-n}f(\xi (t)))\big )_J\\&=(\mathcal {C}_{\xi (t),1})_J\cdots (\mathcal {C}_{\xi (t),n})_J{\dot{\xi }}(t)J^{-n}f(\xi (t)). \end{aligned}$$

This proves that \(C_f(x,t)=\psi _t(x)=\mathcal {C}_{\xi (t),1}(x)\cdots \mathcal {C}_{\xi (t),n}(x){\dot{\xi }}(t)J^{-n}f(\xi (t))\) for all \((x,t)\in E\times T\). As a consequence, we have:

$$\begin{aligned} f_J(x)=(2\pi )^{-n}\int _TC_f(x,t)\,dt\,\,\text { for all }x\in E. \end{aligned}$$

Thanks to Corollary 2.13, in order to complete the proof, it suffices to show that the function \(\Omega _E\rightarrow A\), \(x\mapsto \int _T\psi _t(x)\,dt=\int _TC_f(x,t)\,dt\) is a slice function. Define \(\mathrm {J}:=(J,\ldots ,J)\in ({\mathbb {S}}_A)^n\). Choose \(w=(\alpha _1+J\beta _1,\ldots ,\alpha _n+J\beta _n) \in E\) and \(x=(\alpha _1+L_1\beta _1,\ldots ,\alpha _n+L_n\beta _n) \in \Omega _D\) for some \(L=(L_1,\ldots ,L_n)\in ({\mathbb {S}}_A)^n\). By representation formula (7), we know that

$$\begin{aligned} \textstyle \psi _t(x)=2^{-n}\sum _{K,H \in \mathcal {P}(n)}(-1)^{|K \cap H|}[\mathrm {J}_K,[L_K^{-1},\psi _t(w^{\, c,H})]] \end{aligned}$$

for all \(t\in T\). Hence, it holds:

$$\begin{aligned} \int _T\psi _t(x)\,dt&=\int _T\textstyle \big (2^{-n}\sum _{K,H \in \mathcal {P}(n)}(-1)^{|K \cap H|}[\mathrm {J}_K,[L_K^{-1},\psi _t(w^{\, c,H})]]\big )\,dt\\&=\textstyle 2^{-n}\sum _{K,H \in \mathcal {P}(n)}(-1)^{|K \cap H|}[\mathrm {J}_K,[L_K^{-1},\int _T\psi _t(w^{\, c,H})\,dt]]. \end{aligned}$$

Corollary 2.16 implies that the function \(\Omega _E\rightarrow A\), \(x\mapsto \int _T\psi _t(x)\,dt=\int _TC_f(x,t)\,dt\) is slice, as desired. \(\square \)

Remark 3.40

As a byproduct of the preceding proof, we have that

$$\begin{aligned} C_f(x,t)=\mathcal {C}_{\xi (t),1}(x)\cdots \mathcal {C}_{\xi (t),n}(x){\dot{\xi }}(t)J^{-n}f(\xi (t)) \end{aligned}$$

for all \((x,t)\in E\times T\). \(\square \)

Our next aim is to write \(C_f\) in term of pointwise products of A-valued functions.

For each \((x,y)=((x_1,\ldots ,x_n),(y_1,\ldots ,y_n))\in \Gamma _n(J)\) and for each \(K\in \mathcal {P}(n)\), we denote \(\mathcal {C}(x,y,K):\{1,\ldots ,n\}\rightarrow A\) the function given by

$$\begin{aligned} \mathcal {C}(x,y,K)(h):= \left\{ \begin{array}{ll} (\Delta _{y_h}(x_h))^{-1} &{} \text { if }h\in K, \\ (\Delta _{y_h}(x_h))^{-1}x_h &{} \text { if }h\not \in K. \end{array} \right. \end{aligned}$$
(89)

Theorem 3.41

Let \(f:\Omega _D\rightarrow A\) be a slice regular function. Suppose that \(\mathrm {cl}(\Omega (E))\subset \Omega _D\). Then

$$\begin{aligned} C_f(x,t)=\textstyle \sum _{K\in \mathcal {P}(n)}(-1)^{n-|K|}\big [\mathcal {C}(x,\xi (t),K),\xi _K^c(t){\dot{\xi }}(t)J^{-n}f(\xi (t))\big ] \end{aligned}$$
(90)

for each \((x,t)\in \Omega (E)\times T\).

Proof

Let \(t=(t_1,\ldots ,t_n)\in T\), let \(\psi _t:\Omega (E)\rightarrow A\) be the function \(\psi _t(x):=C_f(x,t)\), let \(y=(y_1,\ldots ,y_n):=(\xi _1(t_1),\ldots ,\xi _n(t_n))\in \partial ^*E\) and let \(a:={\dot{\xi }}(t)J^{-n}f(\xi (t))\). Thanks to Remark 3.40 and Artin’s theorem, given any \(x\in E\), we have that

$$\begin{aligned} \psi _t(x)&=\mathcal {C}_{\xi (t),1}(x)\cdots \mathcal {C}_{\xi (t),n}(x){\dot{\xi }}(t)J^{-n}f(\xi (t))=\frac{(y_1^c-x_1)\cdots (y_n^c-x_n)}{\Delta _{y_1}(x_1)\cdots \Delta _{y_n}(x_n)}\,a\nonumber \\&=\frac{\textstyle \sum _{K\in \mathcal {P}(n)}y_K^c\prod _{h\in \{1,\ldots ,n\}\setminus K}(-x_h)}{\Delta _{y_1}(x_1)\cdots \Delta _{y_n}(x_n)}\,a \end{aligned}$$
(91)
$$\begin{aligned}&=\textstyle \sum _{K\in \mathcal {P}(n)}(-1)^{n-|K|}\big (\prod _{h\in K}\Delta _{y_h}(x_h)^{-1}\big )\nonumber \\&\textstyle \quad \times \big (\prod _{h\in \{1,\ldots ,n\}\setminus K}\Delta _{y_h}(x_h)^{-1}x_h\big )y_K^ca\nonumber \\&=\textstyle \sum _{K\in \mathcal {P}(n)}(-1)^{n-|K|}\big [\mathcal {C}(x,\xi (t),K),y_K^ca]. \end{aligned}$$
(92)

Since \(\psi _t\) is a slice function, in order to complete the proof, it is sufficient to show that the function \(\phi _K:\Omega (E)\rightarrow A\), sending x into \([\mathcal {C}(x,\xi (t),K),y_K^ca]\), is slice for all \(K\in \mathcal {P}(n)\). Indeed, if this is true, then the function \(\sum _{K\in \mathcal {P}(n)}(-1)^{n-|K|}\phi _K\) is slice as well, and by Corollary 2.13 we are done. Let \(K\in \mathcal {P}(n)\). Note that, for each \(h\in \{1,\ldots ,n\}\), the functions \(\Omega (E)\rightarrow A\), \(x=(x_1,\ldots ,x_n)\mapsto \Delta _{y_h}(x_h^c)\) and \(\Omega (E)\rightarrow A\), \(x=(x_1,\ldots ,x_n)\mapsto \Delta _{y_h}(x_h^c)x_h\) are slice preserving and h-reduced. Consequently, an iterated application of Proposition 2.52 implies that \(\phi _K\) is the iterated slice tensor product of \(n+1\) slice functions, which is a slice function as well.

The proof is complete. \(\square \)

Remark 3.42

Repeating the chain of equalities (91) with \(a=1\), we deduce that

$$\begin{aligned} \textstyle \mathcal {C}_{y,1}(x)\cdots \mathcal {C}_{y,n}(x)=\sum _{K\in \mathcal {P}(n)}(-1)^{n-|K|}\big [\mathcal {C}(x,y,K),y_K^c] \end{aligned}$$

for all \((x,y)\in E\times \partial ^*E\). \(\square \)

The associative case

Recall the definition of \(\mathcal {C}_y\) for each \(y\in \partial ^*E\), given in (86).

Definition 3.43

We define the slice Cauchy kernel for E as the function \(C:\Omega (E)\times \partial ^*E\rightarrow A\) given by

$$\begin{aligned} C(x,y):=[\mathcal {C}_y]_\odot (x). \, \end{aligned}$$
(93)

\(\square \)

Remark 3.44

For each \(y\in \partial ^*E\), the function \(\Omega (E)\rightarrow A\), \(x\mapsto C(x,y)\) is slice regular, see Remark 3.38. \(\square \)

Given two continuous functions \(p,q:\partial ^*E\rightarrow A\), if A is associative, then we define

$$\begin{aligned} \int _{\partial ^*E}p(y)dyq(y):=\int _Tp(\xi (t)){\dot{\xi }}(t)q(\xi (t))\,dt. \end{aligned}$$

In the associative setting our Cauchy integral formula assumes a quite familiar form.

Theorem 3.45

Let \(f:\Omega _D\rightarrow A\) be a slice regular function. Suppose that \(\mathrm {cl}(\Omega (E))\subset \Omega _D\). If A is associative then

$$\begin{aligned} f(x)=(2\pi )^{-n}\int _{\partial ^*E}C(x,y)J^{-n}dy f(y)\quad \text { for all }x\in \Omega (E), \end{aligned}$$
(94)

and the slice Cauchy kernel C can be expressed in terms of pointwise products as follows:

$$\begin{aligned} C(x,y)=\textstyle \sum _{K\in \mathcal {P}(n)}(-1)^{n-|K|}[\mathcal {C}(x,y,K),y_K^c] \end{aligned}$$
(95)

for all \((x,y)\in \Omega (E)\times \partial ^*E\).

Proof

Thanks to Remark 3.42, for all \(y\in \partial ^*E\), the slice functions \(\Omega (E)\rightarrow A\), \(x\mapsto C(x,y)\) and \(\Omega (E)\rightarrow A\), \(x\mapsto \sum _{K\in \mathcal {P}(n)}(-1)^{n-|K|}[\mathcal {C}(x,y,K),y_K^c]\) are equal on E. By Corollary 2.13, they coincide on the whole \(\Omega (E)\). This proves (95). Since A is associative, (90) and (95) imply that \(C_f(x,t)=C(x,\xi (t)){\dot{\xi }}(t)J^{-n}f(\xi (t))\) for all \((x,t)\in \Omega (E)\times T\). Consequently, the right hand sides of formulas (88) and (94) coincide. \(\square \)