A strong version of Implicit Function Theorem
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Abstract
We suggest the necessary/sufficient criteria for existence of a (orderbyorder) solution \({y}({x})\) of a functional equation \(F({x},{y})=0\) over a ring. In full generality, the criteria hold in the category of filtered groups, this includes the wide class of modules over (commutative, associative) rings. The classical Implicit Function Theorem and its strengthening obtained by Tougeron and Fisher appear to be (weaker) particular forms of the general criterion. We obtain a special criterion for solvability of equations arising from group actions \(g(w)=w+u\), here u is “small”. As an immediate application we rederive the classical criteria of determinacy, in terms of the tangent space to the orbit. Finally, we prove the Artin–Tougerontype approximation theorem: if a system of \(C^\infty \)equations has a formal solution and the derivative satisfies a Lojasiewicztype condition then the system has a \(C^\infty \)solution.
Keywords
Equations over local rings Equations on groups Functional equations Tougeron’s Implicit Function Theorem Artin’s approximation theorem Tougeron’s approximation theorem Finite determinacyMathematics Subject Classification
47J07 26B10 30D05 39Bxx 65Q201 Introduction
All rings in this paper are commutative, associative, with unit element, of zero characteristic. We use the multivariable notation \({x}=(x_1,\ldots ,x_m)\), \({y}=(y_1,\ldots ,y_n)\).
1.1 General setting and known results
Consider a system of (analytic/formal/\(C^\infty \)/\(C^k\)) equations \(F({x},{y})=0\). The classical Implicit Function Theorem reads: If the matrix of derivatives, Open image in new window , is right invertible (i.e. is of the full rank) then \(F({x},{y})=0\) has a (analytic/formal/etc.) solution.
The condition “ Open image in new window is right invertible” is quite restrictive. For example, the theorem does not ensure a solution of the onevariable equation \(xy=0\) (in the vicinity of (0, 0)) or of \(y^2=0\) (at any point).
Various strengthenings/generalizations of this theorem are known (including the Hensel lemma). For example, the Tougeron Implicit Function Theorem ensures solvability when the matrix Open image in new window is not too degenerate. Denote by Open image in new window the ideal of the maximal minors of this matrix.
Theorem 1.1
([29], [30, p. 56]) Let Open image in new window or Open image in new window (for \(\Bbbk \) a normed field) or Open image in new window . Let \(F({x},{y})\in R^{\oplus p}\), \(p\leqslant n\), and let \(I\subset R\) be a proper ideal. If Open image in new window then there exists a solution \(F({x},{y}({x}))\equiv 0\) such that Open image in new window .
The statement was further strengthened by Fisher, he replaced one of the factors in \(({\mathfrak {a}}_{F'_{{y}}({x},0)})^2\) by the image Open image in new window . (The initial version was for padic rings, we give a more general version relevant to our context.)
Theorem 1.2
In the case of one equation, \(p=1\), this coincides with Tougeron’s result. For \(p>1\), Fisher’s result is stronger. (Note that Open image in new window , and for \(p>1\) the inclusion is in general proper.)
1.2 Overview of results
In this note we prove much stronger solvability criteria. In this introduction we sketch just the main features of the method. The detailed formulation can be found in Sect. 3.1 (Theorem 2.1) and Sect. 5 (Theorem 5.3), the applications are in Sects. 4 and 6.
We weaken the condition on Open image in new window further, to the “weakest possible” condition of “iff” type, so that we get a Strong Implicit Function Theorem.
Our results hold in broader category. It is natural to extend from the classical case of Open image in new window , Open image in new window , Open image in new window to the local Henselian rings (not necessarily regular or Noetherian) over a field. In fact even the ring structure is not necessary, our main result, Theorem 2.1, is true for filtered (not necessarily abelian) groups.
A particular class of equations comes from the group actions Open image in new window . Assume W is a filtered abelian group (e.g. a module over a local ring). To understand how large the orbit is one studies the equation \(g(w)=w+u\). Here \(g\in G\) is an unknown, while \(u\in W\) is “small”. (More precisely, one studies whether the orbit Gw is open in the topology defined by a filtration.) Theorem 2.1, being very general, is of little use here. Rather, we obtain a special version of strong IFT, Sect. 3.1.2.
Usually the main problem is to establish the orderbyorder solution procedure. Thus many of our results are of the form: If Open image in new window then there exists a Cauchy sequence \(\{{y}^{\scriptscriptstyle (n)}({x})\}_n\) such that \(F({x},{y}^{\scriptscriptstyle (n)}({x}))\rightarrow 0\) . (The topology here is induced by a filtration, e.g. by the powers of maximal ideal.)
Once such a result is established, one has a solution in the completion of \(R^{\oplus p}\) by the filtration. Then (if R is noncomplete) one uses the Artintype approximation theorems [18] to establish a solution over R, or at least over the henselization of R.
For the ring \(C^\infty ({\mathbb {R}}^p\!,0)\) and many other important rings the Artin approximation does not hold (in the naive way). Over some rings we can directly ensure a solution, see Sect. 3.4. For \(C^\infty ({\mathbb {R}}^p\!,0)\) we use Theorem 5.3.
1.3 Comments and motivation
Several remarks/explanations are necessary at this point. Recall the simple geometric interpretation. Consider the (germ of) subscheme/subspace Open image in new window . The classical IFT, in the case when (X, 0) and (Y, 0) are smooth, gives a sufficient condition that the germ Open image in new window is smooth and its projection onto (X, 0) is an isomorphism.
Our version of IFT, for arbitrary Henselian germs (X, 0), (Y, 0), gives a necessary and sufficient condition that the germ Open image in new window has an irreducible component whose projection onto (X, 0) is an isomorphism. This can be restated as follows. Consider the natural projection Open image in new window . Usually this projection is not an isomorphism. The solvability of the equation means the weaker property: the existence of the section of \(\pi \), Open image in new window .
To emphasize: as the germ Open image in new window is in general nonsmooth (possibly reducible, nonreduced), the question cannot be simply “linearized” by an automorphism of Open image in new window , i.e. cannot be reduced to the classical IFT by some appropriate change of variables.
A reformulation in terms of commutative algebra reads: Given a ring R over some base ring \(R_X\), e.g. Open image in new window or Open image in new window , etc. Given an ideal \(F=\) \((F_1,\dots ,F_p)\subset R\), a solution of \(F({x},{y})=0\) is a projection Open image in new window whose kernel is precisely F.
The classical approach to construct a solution is the orderbyorder approximation: first solve the part linear in \({y}\) (modulo quadratic terms), then quadratic, cubic, etc. Accordingly, we always present the equation(s) \(F({x},{y})=0\) in the form \({u}+L{y}+H({y})=0\in W\). Here \({u}=F({x},0)\in W\), Open image in new window is a homomorphism of Rmodules (or just of abelian groups); \(H({y})\) denotes the remaining “higher order terms” (a contractive map in the sense of Krull topology), defined in Sect. 2.2.
There exists a subgroup/submodule \(V_1\subseteq V\) such that for any \({v}\in V_1\) the equation \(L({y}{v})+H({y})=0\) has a solution \({y}_{v}\in V_1\) which is “close” to \({v}\) and depends on \({v}\) “differentiably”. We call this a good solution, the precise formulation see in Sect. 2.2. Our criteria answer the question:
Question 1.3
Given Open image in new window , what is the biggest \(V_1\subseteq V\) such that for any \(v\in V_1\) there exists a good solution?
Note that for some equations all solutions are “not good”, cf. Sect. 4.3.
If the number of unknowns equals the number of equations and Open image in new window is nondegenerate, then the classical IFT ensures the unique solution. When Open image in new window is degenerate the solution (if it exists) can be nonunique, as the space Open image in new window can have several irreducible components. However, when L is injective, the solution lying in \(V_1\) is unique! The (non)uniqueness issues are addressed in Sect. 3.3.
We expand \(F({x},{y})=0\) in powers of \({y}\) (i.e. at \({y}=0\)), hence the criteria are formulated in terms of Open image in new window , etc. One can expand at some other point, \({y}=y^{\scriptscriptstyle (0)}({x})\), then the criteria are written in terms of Open image in new window , etc. (For example, Theorem 1.2 is stated in [10] in such a form.) Such an expansion at \(y^{\scriptscriptstyle (0)}({x})\) is helpful if one has a good initial approximation for the solution. The two approaches are obviously equivalent, e.g. by changing the variable \({y}\mapsto {y}y^{\scriptscriptstyle (0)}({x})\). To avoid cumbersome formulas we always expand at \({y}=0\).
In view of our initial result, Sect. 1.2, one might try to weaken the condition on the ideal \({\mathfrak {a}}_{F'_{y}({x},0)}\subseteq J\subset R\) further. It appears that \(J^2=J{\mathfrak {a}}_{F'_{y}({x},0)}\) is almost the “weakest possible” among the conditions stated in terms of ideals only, it cannot be significantly weakened, cf. Sect. 4.2. But this condition is still far from being necessary. The “right” condition (necessary and sufficient) is obtained by replacing the ideals with filtered subgroups. As a bonus we do not need the rings structure anymore, e.g. Theorem 2.1 holds in the generality of (not necessarily abelian) filtered groups.
If the equations \(F({x},{y})=0\) are linear in \({y}\), i.e. Open image in new window , then the obvious sufficient condition for solvability is: the entries of \(F({x},0)\) lie in the ideal \({\mathfrak {a}}_{F'_{{y}}({x},0)}\). While the (tautological) necessary and sufficient condition is: Open image in new window . This condition is much weaker than those of Tougeron and Fisher and is far from being sufficient for nonlinear equations. Therefore as landmarks for our criteria one should consider equations that are nonlinear in \({y}\) .
The Implicit Function Theorem is a fundamental result. In Sect. 4.4 we obtain an immediate corollary to nonbifurcation of multiple polynomial roots under deformations. In Sect. 4.5 we indicate a potential application to the study of smooth curvegerms (lines/arcs) on singular spaces. In Sect. 6.3 we apply a version of strong IFT to groupactions to rederive the classical criteria of finite determinacy.
Further directions in algebra and geometry are: matrix equations, equations on (filtered) groups [3], tactile maps [6], bounds on Artin–Greenberg functions [23, 24], etc. We hope to report on these applications soon.
2 Definitions and notations
2.1 Groups with descending filtration
We always assume that a (not necessarily abelian) group V is filtered by a sequence of normal subgroups \(V\supset V_1\supset V_2\supset \cdots \), Open image in new window . Moreover, we assume that the filtration satisfies the condition \([V_1,V_i]\subseteq V_{i+1}\), similarly to the lower central series of a group. This later condition is trivial when V is an abelian group. If V is complete with respect to \(\{V_i\}\) then the filtration is faithful, i.e. Open image in new window . The filtration induces the Krull topology, the fundamental system of neighborhoods of \(v\in V\) is Open image in new window , or Open image in new window , by normality.
Example 2.1
 (a)
The simplest case is when V is a module over a ring, with filtration defined by the powers of an ideal Open image in new window .
 (b)
Let \((R,{\mathfrak {m}})\) be a local ring with the filtration \(R\supset {\mathfrak {m}}\supset {\mathfrak {m}}^{2}\supset \cdots \). Consider the group of invertible Rmatrices Open image in new window . We get the filtration by normal subgroups Open image in new window .
 (c)Let (X, 0) be the germ of a space (algebraic/formal/analytic etc). Consider the group of its automorphisms Open image in new window . The natural filtration is by the subgroups of automorphisms that are identity up to j’th order. More precisely, denote by \((R_{(X,0)},{\mathfrak {m}})\) the local ring of (germs of) regular functions. Then
2.2 Implicit function equation

\(H({{\mathbbm {1}}}_V)={{\mathbbm {1}}}_W\),

Open image in new window for any \(y\in V_1\) and \(j\in {\mathbb {N}}\).
If V, W are abelian groups, then the implicit function equation is Open image in new window , where Open image in new window , while the higher order \(H({y})\) satisfies \(H(0_V)=0_W\) and Open image in new window for any \(y\in V_1\) and j.
The most common case is when V, W are modules over a (commutative, associative) ring R. Then usually \(L\in \mathrm{Hom}_R(V,W)\). We say that the map Open image in new window is of order \(\geqslant k\) if for any ideal \(J\subset R\) there holds Open image in new window .
Example 2.2
Example 2.3
(Warning) Being of higher order terms can be a restrictive condition. For example, in the equation \(y^2yx+x^a=0\) the monomial \(y^2\) represents the higher order term for the filtration Open image in new window only if \(a\geqslant 3\). Otherwise the condition \(H(V_1)\subseteq L(V_2)\) is not satisfied.
An orderbyorder solution of the equation Open image in new window is a Cauchy sequence \(\{y^{\scriptscriptstyle (n)}\}_{n\geqslant 1}\) with respect to the filtration \(V_\bullet \), i.e. Open image in new window , such that \(L(y^{\scriptscriptstyle (n)})\) \(H(y^{\scriptscriptstyle (n)})L({v})^{1}\in L(V_n)\) . By normality Open image in new window , we can also write the condition as \((y^{\scriptscriptstyle (n+1)}){}^{1}y^{\scriptscriptstyle (n)}\in V_n\) or Open image in new window .

Open image in new window for any \({v}\in V_1\);

\({y}_{{{\mathbbm {1}}}_V}={{\mathbbm {1}}}_V\) and y respects the filtration, \(y(V_i)\subseteq V_i\) (this is a strengthening of continuity);

the map \({y}\) is “differentiable and close to identity”, namely, Open image in new window , where the map Open image in new window is such that Open image in new window for any \(v\in V_1\) and \(j\in {\mathbb {N}}\). Alternatively this condition can be stated as Open image in new window for any Open image in new window . By normality, this is equivalent to Open image in new window .

\(y^{\scriptscriptstyle (n)}_{{{\mathbbm {1}}}_V}={{\mathbbm {1}}}_V\), Open image in new window ,

Open image in new window for all \(n,j\geqslant 1\) and Open image in new window .

\(y^{\scriptscriptstyle (n)}_{{v}}\;y^{\scriptscriptstyle (n+1)}_{{v}}\in V_n\), \(L(y^{\scriptscriptstyle (n)}_{{v}}\;{v})+H(y^{\scriptscriptstyle (n)}_{{v}})\in L(V_n)\),

Open image in new window for all \(n,j\geqslant 1\) and Open image in new window .
2.3 Annihilator of cokernel
By definition, \({\mathfrak {a}}_L W\subseteq L(V)\). In many cases one has a stronger property: Open image in new window for some proper ideal \(J\subsetneq R\).
Example 2.4
Let Open image in new window , \(1<m\leqslant n\), and suppose \({\mathfrak {a}}_L=I_{\max }(L)\), e.g. this holds when \(I_{\max }(L)\) is radical. Then \({\mathfrak {a}}_L W\subseteq J^{m1} L(V)\).
The embedding Open image in new window does not hold only in some degenerate cases. For example, let \(L=\left( {\begin{matrix} f&{}0\\ 0&{}L_1\end{matrix}}\right) \), where \(\det L_1=f\). Then \({\mathfrak {a}}_L=(f)\) and Open image in new window .
3 Main results: criteria of solvability
3.1 General statements
Let Open image in new window be a homorphism of (arbitrary) groups, where V is filtered by normal subgroups as in Sect. 2.1. Consider the equation Open image in new window . See Sect. 2.2 for definitions.
Theorem 2.1
 (i)
If the map Open image in new window represents the “higher order terms”, i.e. Open image in new window for any \(y\in V_1\), \(j\in {\mathbb {N}}\), then there exists a quasigood orderbyorder solution Open image in new window . If moreover L admits a right inverse, i.e. there exists a map Open image in new window such that Open image in new window , \(T(L(V_i))\subseteq V_i\), then there exists a good orderbyorder solution.
 (ii)
Suppose Open image in new window is compatible with the filtration in the sense Open image in new window for some N(j), \(\lim _{j\rightarrow \infty }N(j)=\infty \). If there exists a good orderbyorder solution Open image in new window , then H represents the “higher order terms”, i.e. Open image in new window for any \(y\in V_1\), \(j\in {\mathbb {N}}\).
 (iii)
If V is complete with respect to \(V_\bullet \) and H represents the “higher order terms” then there exists a quasigood solution Open image in new window . If moreover L admits a right inverse, then there exists a good solution.
Proof
(i) First we construct a quasigood orderbyorder solution \(y^{\scriptscriptstyle (n)}\). The procedure is inductive with noncanonical choices. If L is right invertible then all choices are canonical and the solution becomes good.
Note that \(H(V_{i})\subseteq L(V_{i+1})\), cf. Sect. 2.2. Fix some \(v\in V_i\), we construct inductively \(y^{\scriptscriptstyle (n)}\) such that \(y^{\scriptscriptstyle (n+1)}(y^{\scriptscriptstyle (n)})^{1}\in V_{i+n}\) and Open image in new window .
By construction, \(y^{\scriptscriptstyle (n)}\) is a Cauchy sequence, as Open image in new window . And if \(v={{\mathbbm {1}}}_V\) then \(y^{\scriptscriptstyle (n)}={{\mathbbm {1}}}_V\). Moreover, if \(v\in V_i\) then Open image in new window . Thus \(y^{\scriptscriptstyle (n)}\) is a quasigood orderbyorder solution.
(ii) We proceed in two steps. In Step 1 we prove that a good orderbyorder solution is an almost surjective map, its image is dense. In Step 2 we use this auxiliary statement to bound Open image in new window .
Step 1. We prove the following auxiliary statement: If Open image in new window is a good map, i.e. \(y_v=v g(v)\) with Open image in new window , then the image of y is dense in \(V_1\), i.e. for any \(v\in V_1\) there exists a sequence \(v^{\scriptscriptstyle (n)}\in V_1\) such that Open image in new window .
Remark 2.2
To emphasize, this theorem is almost an ‘iff’ statement, thus the assumptions on L, H are the “weakest possible”.
3.1.1 The case of abelian groups
One often needs results of such type for abelian groups, where one solves the equation Open image in new window . We state the corresponding criterion separately.
Corollary 3.3
 (i)
Given abelian groups V, W and a homomorphism Open image in new window . Suppose there exists a decreasing filtration \(V_\bullet \) of V such that for all \({y}\in V_1\) and \(j\geqslant 1\), Open image in new window . Then for any \({v}\in V_1\) there exists a quasigood orderbyorder solution.
 (ii)
If V is complete with respect to \(V_\bullet \) then the conditions Open image in new window imply a quasigood solution of the equation \(L({y}{v})+H({y})=0\).
Remark 3.4
In the classical case of the equation \(F({x},{y})=0\) one requires that the map Open image in new window is right invertible, i.e. surjective. Our criterion demands that \(F'_{y}({x},0)(V_{j+1})\) contains the variation of the higher order terms Open image in new window , here Open image in new window .
3.1.2 A special version for groupaction equations
Given two maps of (not necessarily abelian) groups Open image in new window . Suppose W is filtered by normal subgroups Open image in new window and F(V) contains \({{\mathbbm {1}}}_W\in W\). (Here we do not assume that L is a homomorphism.) Denote by \(\overline{L(V)},\overline{F(V)}\subseteq W\) the closures with respect to the filtration \(W_i\). The following statement is almost tautological, yet highly useful in Sect. 6.
Lemma 3.5
Suppose Open image in new window for any \(j\geqslant k\). If \(W_k\subset \overline{L(V)}\) then \(W_k\subseteq \overline{F(V)}\).
In the abelian case the condition reads Open image in new window .
Proof
Suppose \(W_k\subseteq L(V)\), then Open image in new window for any \(j\geqslant k\). Thus Open image in new window . As F(V) contains \({{\mathbbm {1}}}_W\in W\) we get Open image in new window for any N. Which is precisely \(W_k\subseteq \overline{F(V)}\).
The general case. Let \(N>k\), consider the quotient Open image in new window . Denote the composition maps Open image in new window by \(\pi _N L, \pi _N F\). Then \(W_k\subset \overline{L(V)}\) implies \(\pi _N(W_k)\subseteq \pi _N L(V)\) for any N. By the previous paragraph we get \(\pi _N(W_k)\subseteq \pi _N F(V)\). Thus Open image in new window for any N, which means \(W_k\subseteq \overline{F(V)}\). \(\square \)
3.2 Criteria for modules over the rings
Theorem 2.1 and Corollary 3.3 transform the solvability question into the search for an appropriate filtration \(V_\bullet \). Not much can be said for a general (non)abelian group. However, our criterion simplifies for modules over a ring: it is enough to find just the first submodule \(V_1\subset V\) and an ideal.
Example 3.6
Such an approximation holds e.g. for R a subring of one of the quotients Open image in new window .
Corollary 3.7
 (a)
If the equation Open image in new window admits a good orderbyorder solution for the filtration \(\{V_i=J^{i1}V_1\}\) then Open image in new window .
 (b)
If Open image in new window then Open image in new window admits a quasigood orderbyorder solution for the filtration \(\{V_i=J^{i1}V_1\}\). (If L is right invertible then there exists a good orderbyorder solution.)
Proof
(a) By Theorem 2.1, the existence of a good solution implies Open image in new window and hence Open image in new window .
(b) For any \(t\in \Bbbk \), \(\Delta \in V_1\) we have Open image in new window . Thus Open image in new window for \(t\in \Bbbk \). Then Open image in new window and \(H_2({y},t\Delta )\) Open image in new window . Thus Open image in new window and Open image in new window . Thus Open image in new window is implied by Open image in new window . Now invoke Corollary 3.3 for the filtration \(\{V_i=J^{i1}V_1\}\). \(\square \)
Corollary 3.7 reduces the question (for modules over a ring) to the search for an appropriate submodule \(V_1\subset V\). The simplest submodule is Open image in new window for some ideal \(J\subset R\).
Corollary 3.8
Suppose H(y) has the linear approximation as in equation (4), and moreover H(y) is of order \(k\geqslant 2\), i.e. Open image in new window . If Open image in new window then there exists a quasigood orderbyorder solution Open image in new window with respect to the filtration Open image in new window .
(Proof: Apply Corollary 3.7 for the filtration Open image in new window .)
Example 3.9
 (a)
Consider the annihilator of cokernel ideal Open image in new window , cf. Sect. 2.3. If Open image in new window then there exists a good (orderbyorder) solution Open image in new window . Also, a bit weaker form: if Open image in new window then there exists a good (orderbyorder) solution Open image in new window .
In the lowest order case, \(k=2\), we get a sufficient condition for the orderbyorder solvability: Open image in new window . This condition is weaker than Tougeron’s and Fisher’s conditions, so even this criterion is stronger.
 (b)
Quite often Open image in new window , cf. Sect. 2.3. Then we get a stronger statement: if Open image in new window then there exists a good (orderbyorder) solution Open image in new window .
3.2.1 Ideals that satisfy \(J^2\subseteq J{\mathfrak {a}}_L\)
(These are important in view of Example 3.9.) Consider the set \(\mathfrak {J}\) of all ideals satisfying \(J^2\subseteq J{\mathfrak {a}}_L\). This is an inductive set, i.e. for any increasing sequence \(J_1\subseteq J_2\subseteq \cdots \) the union Open image in new window is an ideal such that \(J^2\subseteq J{\mathfrak {a}}_L\). (If Open image in new window then \(f,g\in J_k\) for some \(k<\infty \), thus \(fg,f+g\in J_k\).) Therefore in \(\mathfrak {J}\) there exist(s) ideal(s) that is/are maximal by inclusion.
Lemma 3.10
 (i)
\({\mathfrak {a}}_L\subseteq J\). If the ideal J is finitely generated then \(J\subseteq \overline{{\mathfrak {a}}_{L}}\). (Here \(\overline{{\mathfrak {a}}_{L}}\) is the integral closure.)
 (ii)
If \({\mathfrak {a}}_L\) is radical then \(J={\mathfrak {a}}_L\). If R is integrally closed and \({\mathfrak {a}}_L\) is principal, generated by a nonzero divisor, then \(J={\mathfrak {a}}_L\).
Proof
(i) If \(J^2\subseteq J{\mathfrak {a}}_L\) then obviously the inclusion is satisfied by the ideal \(J+{\mathfrak {a}}_L\) as well. As J is the largest with this property, \({\mathfrak {a}}_L\subseteq J\). For the second part, note that \({\mathfrak {a}}_L\) is a reduction of J, see [17, Definition 1.2.1], thus \(J\subseteq \overline{{\mathfrak {a}}_{L}}\) by [17, Corollary 1.2.5].
(ii) If \(J^2\subseteq J{\mathfrak {a}}_L\) then in particular \(J^2\subset {\mathfrak {a}}_L\). Then, \({\mathfrak {a}}_L\) being radical, we get \(J\subseteq {\mathfrak {a}}_L\). Hence together with (i) we get \(J={\mathfrak {a}}_L\). The second part follows from [17, Proposition 1.5.2]: in our case \(\overline{{\mathfrak {a}}_{L}}={\mathfrak {a}}_L\). \(\square \)
Example 3.11
In many cases \({\mathfrak {a}}_L\subsetneq J\subsetneq \overline{{\mathfrak {a}}_{L}}\) and a maximal by inclusion ideal J is nonunique. For example, let Open image in new window and Open image in new window . Then \({\mathfrak {a}}_L=(x^p,y^p,z^p)\) while \(\overline{{\mathfrak {a}}_{L}}={\mathfrak {m}}^p\). Define \(J_z=((x,y)^p,z^p)\), \(J_y=((x,z)^p,y^p)\), \(J_x=((y,z)^p,x^p)\). By a direct check, each of them satisfies \(J^2=J{\mathfrak {a}}_L\). But there is no bigger ideal J that contains say \(J_x+J_y\) and satisfies \(J^2=J{\mathfrak {a}}_L\). Indeed, suppose \(y^{pi}z^{i}\in J\) and \(x^{pj}z^{j}\in J\), for some i, j satisfying \(i+j<p\). Then \(J{\mathfrak {a}}_L=J^2\ni x^{pj}y^{pi}z^{i+j}\), in particular \(x^{pj}y^{pi}z^{i+j} \in {\mathfrak {a}}_L\), \(i+j<p\), contradicting the definition of \({\mathfrak {a}}_L\). Thus, in this case there are at least three distinct maximal by inclusion ideals.
3.3 (Non)Uniqueness
The classical Implicit Function Theorem ensures the uniqueness of solution, provided \(F'_{y}(0,0)\) is injective. In our case the injectivity ensures that the solution is “eventually unique” in the following sense.
Proposition 3.12
Given two orderbyordersolutions \(y^{\scriptscriptstyle (n)}_1\!, y^{\scriptscriptstyle (n)}_2\) of the equation Open image in new window . Suppose Open image in new window and L is injective. Then for any n, Open image in new window .
Proof
By the assumption Open image in new window . Suppose the statement holds for \(j=1,\dots ,n1\). As both \(y^{\scriptscriptstyle (n)}_i\) are Cauchy sequences, we get Open image in new window . We shall prove that in fact Open image in new window .
Remark 3.13
The assumption \(y^{\scriptscriptstyle (1)}_1\!, y^{\scriptscriptstyle (1)}_2\in V_1\) is important. One might seek for a condition in terms of \({v}\) and L only, then it is natural to ask whether \({v}\) belongs to a small enough subgroup of V. For example, in the case of modules, \(v\in JV\), for some small enough ideal \(J\subset R\). This does not suffice as one sees already in the example of one equation in one variable \((yx^a)(y+x^b)=0\). Suppose \(a<b\), then \({\mathfrak {a}}_L=(x^a)\), while \(v\in (x^{a+b})\). By taking \(b\gg a\) the ideal \((x)^{a+b}\) can be made arbitrarily small as compared to \({\mathfrak {a}}_L\). Yet, there is no uniqueness.
Remark 3.14
If L is noninjective then there can be no uniqueness. Even the images \(L(y^{\scriptscriptstyle (n)})\) of an orderbyordersolution are not “eventually unique”. As the simplest example consider the equation \(y^2_1+y^2_2y_1+v=0\), where Open image in new window . We have a family of solutions \(y_1=2(v+y^2_2)/\bigl (1+\sqrt{14(v+y^2_2)}\bigr )\), here \(y_2\) is a parameter. By taking \(y_2\in (v^j)\) these solutions can be made arbitrarily close one to the other (in particular they all lie in \(V_1\)), yet \(L(y_1,y_2)\) is different for different \(y_2\).
3.4 A criterion for exact solutions
The criteria of Sect. 3.1 provide orderbyorder solutions, alternatively, solutions in the completion of V by \(V_\bullet \), i.e. the formal solutions. Recall the Artin approximation property: if a finite system of polynomial equations over R has a solution over \(\widehat{R}\) then it has a solution over R [1, 2]. Many rings have this approximation property, e.g. excellent Henselian rings (in particular complete rings, analytic rings), cf. [16].
In our case we have more general rings and more general class of equations. Thus we give a criterion for exact (and not just orderbyorder) solutions.
Fix some proper ideal \(J\subsetneq R\). The pair (R, J) is said to satisfy the (classical) Implicit Function Theorem, denote this by Open image in new window , if for any surjective morphism of free Rmodules of finite ranks Open image in new window , any Open image in new window and any “higher order term” Open image in new window , the equation Open image in new window has a good solution. Note that if R satisfies Open image in new window then for any ideal \(J_1\subseteq J\) the ring satisfies Open image in new window as well.
Example 3.15
Let \((R,{\mathfrak {m}})\) be any local Henselian ring over a field \(\Bbbk \). For example, the ring of formal power series Open image in new window , the ring of analytical power series Open image in new window (for \(\Bbbk \)normed), the ring of smooth functions Open image in new window or the ring of ptimes differentiable functions Open image in new window . Then \((R,{\mathfrak {m}})\) satisfies \(\mathrm{cIFT}_{\mathfrak {m}}\).
The rings Open image in new window , Open image in new window do not satisfy \(\mathrm{cIFT}_{\mathfrak {m}}\), e.g. the equation \(y^2+y=x^2\) is not solvable over these rings.
We say that (R, J) satisfies the Implicit Function Theorem with unit linear part, denote this by Open image in new window , if the system of equations \({y}{v}+H({y})=0\) has a good solution, Open image in new window , for any higher order terms H.
This system is a particular case of the classical implicit function equations. Therefore the Henselian rings (over a field) of Example 3.15 satisfy \(\mathrm{IFT}_{{\mathfrak {m}}\!,{\mathbbm {1}}}\). Note that the condition Open image in new window is weaker than Open image in new window . For example, Open image in new window is satisfied by Open image in new window , Open image in new window for \(J=({x})\). More generally, one can take as \(\Bbbk \) any ring and as R a Henselian algebra over \(\Bbbk \).
Proposition 3.16
Given a finitely generated Rmodule V and two maps Open image in new window . Suppose \(L\in \mathrm{Hom}_R(V,W)\), while H satisfies Open image in new window , here \(\{\xi _i\}\) are some generators of V, while \(h_i(\{y_j\})\) are of order \(\geqslant 2\). Suppose Open image in new window holds for an ideal \(J\subsetneq R\). Then the equation Open image in new window has a solution Open image in new window .
Note that here R is not necessarily over a field, e.g. R can be Open image in new window or Open image in new window . Being of order \(\geqslant 2\) means that \(h_i(J)\subseteq J^2h(R)\) for any ideal \(J\subseteq R\).
Proof
Expand \(v=\sum _i v_i\xi _i\), \(y=\sum _i y_i\xi _i\), then the equation reads Open image in new window . Thus it is enough to solve the finite system of equations Open image in new window . As Open image in new window holds in our situation we get the solution. \(\square \)
Corollary 3.17
 (i)
If Open image in new window then for any Open image in new window there exists a solution.
 (ii)
If Open image in new window then for any Open image in new window there exists a solution.
 (iii)
If Open image in new window and Open image in new window then for any Open image in new window there exists a solution.
Example 3.18
Let \((R,{\mathfrak {m}})\) be a local Henselian ring over a field. Take \(J={\mathfrak {a}}_L\), then the corollary implies Tougeron’s and Fisher’s theorems. As mentioned in the introduction, if one takes J the maximal possible that satisfies Open image in new window then one gets the strengthening of Tougeron’s and Fisher’s theorems.
But the corollary is useful for more general rings, e.g. if in equation (2) the term \(p(x_1,x_2)\) has integral coefficients then we get a solution over Open image in new window .
4 Examples, remarks and applications
4.1 Comparison to Fisher’s and Tougeron’s theorems
The condition Open image in new window (cf. Corollary 3.17) is a weakening of the condition \(J\subseteq {\mathfrak {a}}_{F'_{{y}}({x},0)}\).
Example 3.1
Let Open image in new window , where \(\Bbbk \) is some base ring, take Open image in new window . (If \(\Bbbk \) is a field then \({\mathfrak {m}}\) is the maximal ideal.) Consider the equation Open image in new window , compare this to (2). Here \(H({y},{x})\) represents the higher order terms, it is at least quadratic in \(y_1,y_2\), while \(p({x})\in R\). In this case, Open image in new window and \(I_{\max }(L)={\mathfrak {a}}_{L}=(x^k_1,x^k_2)\subset R\). Thus \(({\mathfrak {a}}_L)^2=(x^{2k}_1,x^{k}_1x^{k}_2,x^{2k}_2)\). Thus to apply Tougeron’s and Fisher’s theorems we have to assume that Open image in new window .
On the other hand, by a direct check, the ideal Open image in new window satisfies Open image in new window . Therefore Corollary 3.17 implies: if \(p({x})\in {\mathfrak {m}}^{2k+1}\) then the equation has a solution. For \(\Bbbk \) an algebraically closed field we get a better criterion: if \(p({x})\in {\mathfrak {m}}^{2k}\) then the equation has a solution.
Note that to write down an explicit solution is not a trivial task even in the particular case of (2).
Further, if \(\Bbbk \) is not a field then we get the solvability of a “Diophantine type” equation. For example, for \(\Bbbk ={\mathbb {Z}}\) and \(H(y,{x}),p({x})\) defined over \({\mathbb {Z}}\), we get the criterion of solvability over Open image in new window . Note that even for the equation \(y^n+yx^k+x^N=0\) the solvability over Open image in new window is not totally obvious.
Therefore, even in the case of just one equation, the condition Open image in new window strengthens the versions of Tougeron and Fisher.
4.2 Comparison of the condition Open image in new window to \(H(V_1)\subseteq J L(V_1)\)
(cf. Corollary 3.7) It is simpler to check the ideals Open image in new window than to look for a submodule satisfying the needed property. But the “idealtype” criterion is in general weaker than the criterion via \(V_1\).
Example 4.2
Of course, the general criterion of Corollary 3.7 suffices here. (One starts from \(V_1=\left( {\begin{matrix}x_1 R\\ x_2 R\end{matrix}}\right) \) and \(J=(x_1,x_2)\).)
This is a good place to see in a nutshell why no weakening of Open image in new window in the form of some condition on ideals is possible.
Example 4.3
Example 4.4
Remark 4.5

Suppose L is blockdiagonal. What are the conditions on H so that we can choose Open image in new window ?

Formulate some similar statements for L upperblocktriangular v.s. \(V_1\) an appropriate extension of modules.
4.3 Equations whose solutions are not good
Often the “simple” and “most natural” solutions are not good (not even quasigood) in our sense, moreover the (quasi)good solutions do not exist at all.
Example 4.6
Consider the equation \(y^2=p(x)\) over Open image in new window . Here \(L=0\), while \(H(y)\ne 0\). Corollary 3.7 claims that there are no good solutions. Explicitly, there does not exist a submodule \(\{0\}\ne V_1\subset R\) such that for any \(p(x)\in V_1\) there exists a solution Open image in new window good in the sense of Sect. 2.2. This can be seen directly, if \(V_1\ne \{0\}\) then Open image in new window for \(N\gg 1\), and \(y^2=x^{2N+1}\) has no solutions in R.
Of course, by a direct check, for any p(x) of even order there are solutions. But these solutions are not good.
Example 4.7
Consider the equation Open image in new window over Open image in new window , \({x}=(x_1,\ldots ,x_n)\), \(n>1\). Assume that \(g_1({x}), g_2({x})\) are generic enough, in particular Open image in new window . Then the equation cannot be presented in the form Open image in new window , so it has no quasigood solutions. (Even its linear part is nonsolvable, though the equation has two obvious solutions.) This happens because an arbitrarily small deformation of the free term, Open image in new window , will bring an equation with no solutions in Open image in new window . (In the case \(g_1({x}),g_2({x})\in C^\infty ({\mathbb {R}}^p,0)\) even a deformation by a flat function will lead to an equation with no solutions.)
4.4 An application: bifurcations of polynomial roots
Fix a polynomial \(p(y)=\sum ^d_{i=0}a_i y^i\). Suppose for a fixed tuple of the coefficients \((a_0,\ldots ,a_d)\) the polynomial has only simple roots (of multiplicity one). Then under small deformations of coefficients the roots deform smoothly. The multiple roots cause bifurcations in general. Our results provide a sufficient condition that a particular root deforms (smoothly/analytically/etc.) under the change of parameters. More precisely, starting from the initial ring R consider an extension S of R by one local variable, e.g. Open image in new window or Open image in new window , etc. Present the family of equations in the form Open image in new window . We say that a root \(y_0\) of the initial equation deforms (smoothly/analytically/etc.) if there exists a root \(y(t)\in S\) such that \(y(0)=y_0\).
To formulate the criterion we shift the variables \(y\mapsto y+y_0\), so that the (new) root of the initial equation is \(y=0\).
Corollary 4.8
 (i)
(Tougeron) If \(a_0(t)\in (ta_1^2(t))\) then the root \(y=0\) of the initial equation deforms with t.
 (ii)
(Belitskii–Kerner) If \(a_0(t)\in (t a_1(t))\) and Open image in new window for any \(i\geqslant 2\) then the root \(y=0\) of the initial equation deforms with t.
(Note that if \(a_0(t)\in (ta_1^2(t))\) then all assumptions of part two are satisfied.) To check this statement it is enough to put \(v=a_0/a_1\) and Open image in new window .
Example 4.9

(Tougeron’s part) If \(\det A_t \in (t(\mathrm{trace}\,A^\vee _t)^2)\) then the eigenvalue deforms smoothly.

(Belitskii–Kerners’s part) If Open image in new window then the eigenvalue deforms smoothly.
4.5 A possible application: smooth curvegerms on singular spaces
Let \((X,0)\subset (\Bbbk ^n\!,0)\) be a germ (algebraic/analytic/formal) of a singular space. The smooth curvegerms lying on (X, 0) are an important subject, often used in the theory of arc spaces [8]. The first question is whether (X, 0) admits at least one smooth curvegerm, [13, 14, 15].
From the IFT point of view this question reads (for simplicity we work over Open image in new window ): Can a given system of equations be augmented by another system so that the total system Open image in new window has a onedimensional power series solutions? For example Open image in new window , \(F(x_1,x_2(x_1),\ldots ,x_n(x_1))\equiv 0\) ? The strong IFT seems to lead to some results on the existence/properties of families of such curves.
5 An approximation theorem of Artin–Tougeron type
A formal solution of this equation is a formal series Open image in new window such that \(\widehat{F}({x},\widehat{y}({x}))\equiv 0\), where \(\widehat{F}\) is the (formal) Taylor expansion at zero of the map F. In general this solution does not converge off the origin. Two classical results relate it to the “ordinary” solution.
Theorem 5.1
 (i)
For every \(r\in {\mathbb {N}}\) there exists an analytic solution whose r’th jet coincides with the r’th jet of \(\widehat{y}({x})\) [1].
 (ii)
There exists a \(C^\infty \)solution \({y}({x})\) whose Taylor series at the origin is precisely \(\widehat{y}({x})\) and such that for any \(r\in {\mathbb {N}}\) there exists an analytic solution which is rhomotopic to \({y}({x})\) [31].
(Recall that two solutions \({y}_0({x}), {y}_1({x})\) are rhomotopic if there exists a \(C^\infty \)family of solutions \({y}({x},t)\) such that \({y}_0({x})={y}({x},0)\), \({y}_1({x})={y}({x},1)\) and \({y}({x},t){y}_0({x})\) is rflat at the origin.)
What if the equation \(F({x},{y})=0\) is not analytic but only of \(C^\infty \)type? Does the existence of a formal solution for the completion \(\widehat{F}({x},{y})=0\) imply the existence of a \(C^\infty \)solution? The naive generalization of Artin/Tougeron’s theorems does not hold.
Example 5.2
In this example the coefficient of y(x), i.e. the function \(\tau ^2\), is flat at zero. In other words, the ideal \({\mathfrak {a}}_{F'_{y}({x},{y}_0)}\) is too small and Open image in new window .
The following statement supplements our previous results and extends Tougeron’s theorem to \(C^\infty \)equations. Let \(R=C^\infty ({\mathbb {R}}^m\!,0)\) with the maximal ideal \({\mathfrak {m}}\subset R\). Suppose the equation \(F({x},{y})=0\) has a formal solution \(\widehat{{y}}_0\). By Borel’s lemma [25] we can choose a \(C^\infty \)map \({y}_0\) whose completion is \(\widehat{{y}}_0\), thus \(F({x},{y}_0)\) is a vector of flat functions.
Theorem 5.3
Suppose the equation \(F({x},{y})=0\) has a formal solution and there holds Open image in new window . Then there exists a \(C^\infty \)map, Open image in new window whose Taylor series at the origin is precisely \(\widehat{y}_0\) and such that \(F({x},{y}({x}))\equiv 0\).
Proof
Remark 5.4
 (a)
The assumption of the theorem can be stated as: Every function flat at the origin is divisible by Open image in new window . In particular this implies that Open image in new window is nondegenerate in some punctured neighborhood of the origin \(0\in ({\mathbb {R}}^m\!,0)\). Note that \({y}_0\) is defined up to a flat function, but the assumption does not depend on this choice.
 (b)Recall that a function \(g({x})\) is said to satisfy the Lojasiewicz condition (at the origin) if there exist constants \(C>0\) and \(\delta >0\) such that for any point \({x}\in ({\mathbb {R}}^m\!,0)\): Open image in new window . As is proved, e.g. in [30, Section V.4], \(g({x})\) satisfies the Lojasiewicz condition at the origin iff Open image in new window . Thus the assumption of the theorem can be stated in the form
 (c)
A similar statement can be proved for \(C^k({\mathbb {R}}^p\!,0)\)functions, but then the solution is in general only in the \(C^{k2\delta }\)class.
6 Openness of group orbits and applications to the finite determinacy
Given a module W over some base ring \(\Bbbk \) (we assume \(\Bbbk \supseteq {\mathbb {Q}}\)) with a decreasing filtration \(\{W_i\}\). Consider the group of all \(\Bbbk \)linear invertible maps that preserve the filtration Open image in new window . Fix some subgroup Open image in new window and let \(G^0\subseteq G\) be the unipotent subgroup, Sect. 6.1.1. Fix some element \(w\in W\), consider the germ of its \(G^0\)orbit \((G^0w,w)\) and the tangent space to this germ \(T_{(G^0w,w)}\), Sect. 6.1.2. (Note that the existence of \(T_{(G^0w,w)}\) places some restrictions on G, see (8).)
Theorem 6.1
If \(W_k\subseteq \overline{T_{(G^0w,w)}}\) then \(w+W_k\subseteq \overline{G^0w}\).
Here Open image in new window denotes the closure with respect to the filtration \(W_\bullet \). Thus the statement is of the orderbyordertype. In particular, in the proof we can assume that W is \(W_\bullet \)complete. The proof is given in Sect. 6.2, after some preparations in Sect. 6.1. Some immediate applications to the finite determinacy are given in Sect. 6.3.
6.1 Preparations
6.1.1 The unipotent subgroup \(G^0\)
Example 6.2
 (a)
Let \(G=\mathcal {R}\) be the group of local coordinate changes \({x}\mapsto \phi ({x})\). They act on the elements of the ring by \(f({x})\mapsto \phi ^*(f({x}))=f(\phi ({x}))\). For the filtration \(\{{\mathfrak {m}}^{j}\}\) the group \(G^0\) consists of the changes of the form \({x}\mapsto {x}+h({x})\), where \(h({x})\in {\mathfrak {m}}^{2}\).
 (b)More generally, consider the group of automorphisms of a module Open image in new window Open image in new window acting by Open image in new window , where \(\phi \in \mathcal {R}\), while \(U({x})\) is an invertible matrix over R. Then
 (c)
Note that \(G^0\) depends essentially on the filtration. In the previous examples we could take the filtration by the powers of some other ideal \(\{J^i\}\) or just by a decreasing sequence of ideals.
6.1.2 Logarithm, exponent and the tangent space
As is mentioned after Theorem 6.1 we can pass to the completion of the module \({\widehat{W}}\) with respect to the Open image in new window filtration. Accordingly we have Open image in new window , the completions of Open image in new window .
Definition 6.3
The tangent space to \({\widehat{G}}{}^0\) at the unit element is the \(\Bbbk \)module \(T_{\scriptscriptstyle {\widehat{G}}{}^0}=\ln {\widehat{G}}{}^0\subseteq \mathrm{End}_\Bbbk ({\widehat{W}})\).
Lemma 6.4
The image \(\exp T_{\scriptscriptstyle {\widehat{G}}{}^0}={\widehat{G}}{}^0\) and the maps \(T_{\scriptscriptstyle {\widehat{G}}{}^0}\underset{\scriptscriptstyle \ln }{\mathop {\rightleftarrows }\limits ^{\scriptscriptstyle \exp }}{\widehat{G}}{}^0\) are mutual inverses.
Proof
Let \(\xi \in T_{\scriptscriptstyle {\widehat{G}}{}^0}\), then \(\xi =\ln g\) for some \(g\in {\widehat{G}}{}^0\). Thus \(\exp \xi =\exp \ln g=g\in {\widehat{G}}{}^0\). The maps \(\ln \) and \(\exp \) are mutual inverses as they are defined by the same Taylor series as the classical functions. \(\square \)
6.1.3 The relevant properties of the exponent and variation operator
The j’th stabilizer of \(w\in W\) is the subgroup Open image in new window . For any \(g\in G^0\) and \(w\in W\) define the variation operator \(\Delta _w(g)=gww\).
Lemma 6.5
The restriction Open image in new window , \(j\geqslant 1\), is a homomorphism of groups.
Proof
Lemma 6.6
 (i)
\(\pi _j(\exp \xi )\in \pi _j(\mathrm{St}_j(w))\) iff \(\pi _j(\xi w)=0\in \pi _j(W)\).
 (ii)
If \(\pi _j (\exp \xi )\in \pi _{j}( \mathrm{St}_{j}(w))\) then \(\pi _{j+1}(\Delta _w(\exp \xi ))=\pi _{j+1}(\xi w)\).
Proof
(i) (\(\Rightarrow \)) As the stabilizer is a group, \(\pi _j(\exp t\xi w)=\pi _j (w)\) for all \(t\in {\mathbb {Z}}\). The left hand side of this equation is a polynomial in t because \(\xi \) is nilpotent. As \(\mathrm{char}\,\Bbbk =0\) and \(\Bbbk \supseteq {\mathbb {Q}}\), the equality holds for all \(t\in \Bbbk \). But this implies \(\pi _j (\xi w)=0\).
(\(\Leftarrow \)) If \(\pi _j(\xi w)=0\) then \(\pi _j(\xi ^k w)=0\), thus \(\pi _j (\exp \xi )\in \pi _j (\mathrm{St}_j(w))\).
(ii) The function \(h(t)=\pi _{j}(\Delta _w(\exp t\xi ))\) is a polynomial in t. By Lemma 6.5, it is additive. Thus \(h(t)=tc\) where \(c=h(1)=\pi _j (\exp \xi ww)=\pi _{j}(\xi w)\). \(\square \)
6.2 Proof of Theorem 6.1
6.3 An application to finite determinacy
Let \((R,{\mathfrak {m}})\) be as in Example 3.6. Below we describe several scenarios (the module and the group action), in each case it is enough to write down the corresponding tangent space(s).
Example 6.7

If Open image in new window then f is k\({\mathcal {R}}^0\)determined.

If Open image in new window then f is k\({{\mathcal {K}}}^0\)determined.
Example 6.8
More generally, let \(W=R^{\oplus p}\) with the filtration \({\mathfrak {m}}^{j} W\). The contact group action can be written as Open image in new window , where Open image in new window , Open image in new window . For the unipotent part \({{\mathcal {K}}}^0\) one has: Open image in new window , Open image in new window . Then Open image in new window .
Example 6.9
Example 6.10
When the hypersurface singularity Open image in new window is nonisolated, the tangent space Open image in new window does not contain \({\mathfrak {m}}^{k}\) for any k. Thus the filtration \(\{{\mathfrak {m}}^{j}\}\) is irrelevant. It is natural to consider only the deformations preserving the singular locus. More precisely, for the ideal \(\mathrm{Jac}_f+(f)\) consider the following saturation. Take the primary decomposition \(\bigcap _i I_i\) and apply the procedure: if \(\sqrt{I_i}\supsetneq \sqrt{I_j}\) then erase \(I_i\) in this decomposition. Eventually one gets a saturated version \((\mathrm{Jac}_f+(f))^\mathrm{sat}\), geometrically this corresponds to removing the embedded components of lower dimension. Then one can consider either of the filtrations Open image in new window or Open image in new window , here Open image in new window . The later filtration has been studied in [22, 26, 27].
In both cases one defines Open image in new window and considers the corresponding subgroup Open image in new window of \({\mathcal {R}}^0\). In both cases one has: If \(\mathrm{Der}_{W_1}(R)(f)\supseteq {\mathfrak {m}}^{k} W_1\) then f is k determined for deformations inside \(W_1\).
Notes
Acknowledgments
We thank Jacek Bochnak, Herwig Hauser, Dorin Popescu, Jesús M. Ruiz, Eugenii Shustin, and Sergei Yakovenko for the attention and valuable suggestions. We also thank two referees, their numerous remarks have greatly improved the exposition.
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