A note on polydegree $(n,1)$ rational inner functions, slice matrices, and singularities

We analyze certain compositions of rational inner functions in the unit polydisk $\mathbb{D}^{d}$ with polydegree $(n,1)$, $n\in \mathbb{N}^{d-1}$, and isolated singularities in $\mathbb{T}^d$. Provided an irreducibility condition is met, such a composition is shown to be a rational inner function with singularities in precisely the same location as those of the initial function, and with quantitatively controlled properties. As an application, we answer a $d$-dimensional version of a question posed in \cite{BPS22} in the affirmative.


Introduction
Background. This note is concerned with certain bounded holomorphic functions on the unit polydisk in C d , here and throughout, d ∈ N. By Fatou's theorem for polydisks (see e.g. [20,Chapter 3]), any bounded holomorphic function φ : D d → C has non-tangential boundary values φ * (ζ) = ∠ lim D d ∋z→ζ φ(z) at almost every point ζ ∈ T d . If these boundary values satisfy |φ * (ζ)| = 1 for almost every ζ ∈ D d , we say that φ is an inner function.
Inner functions of the form φ = q/p, where q, p ∈ C[z 1 , . . . , z d ] and p has no zeros in D d , are called rational inner functions (RIFs). In one variable, RIFs are precisely the finite Blaschke products in the unit disk D. Blaschke products play a central role in function theory, see for instance [11] for an overview of the very rich theory of these functions. In two and more variables, RIFs are a concrete class of bounded holomorphic functions that is amenable to detailed study [20,SOLA Chapter 5], and appears naturally in several setting, for instance in connections with interpolation problems [1].
A classical result of Rudin and Stout (see [20,Chapter 5]) states that any RIF in D d admits a representation of the form where a ∈ R, m = (m 1 , . . . , m d ) ∈ N d , andp is the reflection of a polynomial p with no zeros in D d known as a stable polynomial. The reflection polynomial is defined as The vector (n 1 , . . . , n d ) is referred to as the polydegree of p; each n j = deg z j (p) is the degree of p in the variable z j . In this note, we shall strip out monomial factors and consider RIFs φ = e iap /p; this simplifies formulas and is not material for the problem we study. RIFs as well as more general bounded rational functions in two or more variables have been considered by a number of authors in recent years, often in connection with stable polynomials, representation formulas, and operator-theoretic problems. We cannot give a full overview here, but a sampler of related work might include papers of Anderson, Dritschel, and Rovnyak [3]; Ball, Sadosky, and Vinnikov [4]; Knese [13,14,15], and Kollár [17].
A series of recent papers with Bickel and Pascoe [7,8,9]; Bickel, Knese, and Pascoe [10]; and Tully-Doyle [21] deal with aspects of RIF theory that are particular to dimensions d ≥ 2. Namely, unlike in one dimension, RIFs in two or more variables can have singularities on the d-torus, arising at points ζ ∈ T d where p(ζ) = 0 andp(ζ) = 0 vanish without having common factors that cancel out. A d-dimensional example (see [15,Section 5] and [9, Example 2.5]) is given by which has a singularity at (1, . . . , One would like to describe RIF singularities in detail, and there are different ways of doing this. The papers [7,8,9], as well as [10], investigate for which p ≥ 1 the partial derivative of a RIF has ∂φ ∂z d ∈ L p (T d ). Roughly speaking, the smaller the maximal p for which integrability holds, the stronger the singularity of φ; for the example (1.2), the maximal integrability index is p = 1 2 (d + 1); see [9] and [10] for comprehensive discussions. The paper [7] and the work of Bergqvist [5] also consider other notions of derivative integrability corresponding to norms of Dirichlet type.
Overview of results. The purpose of this short note is to present some straight-forward observations regarding d-variable RIFs of polydegree (n, 1), n = (n 1 , . . . , n d−1 ) ∈ N d−1 , and their singularities. This restricted class of functions is often singled as more amenable to analysis, see for instance [13,9,6]. Ifζ = (ζ 1 , . . . ζ d−1 ) ∈ T d−1 is kept fixed and φ =p/p is a RIF in D d , the resulting one-variable function φζ(z d ) is either a Möbius transformation mapping the unit disk onto itself, or else is a unimodular constant. By encoding this fact in a 2 × 2 matrixvalued function ofζ, and expressing the determinant of this matrix in terms ofζ-polynomials extracted from p andp, we are able to read off certain geometric characteristics of such φ.
This allows us to exhibit d-variable RIFs with prescribed singularity types, and hence derivative integrability properties, while keeping the z d -degree of the resulting functions equal to 1. As a specific application, we are able to answer a stronger version of [9,Question 3] in the affirmative.

Preliminaries
Polydegree (n, 1) RIFs and their singularities. Let p be an irreducible stable polynomial in D d , the latter meaning that Z(p) = {z ∈ C d : p(z) = 0} does not intersect D d . We assume throughout that p has polydegree (n, 1) where n = (n 1 , . . . , n d ) ∈ N d−1 and that p is atoral, which in this context means that p andp share no common factor, see [9, Section 1.2]. Then we can decompose p as a sum where p 1 (ẑ) and p 2 (ẑ) are in C[z 1 , . . . , z d−1 ], and similarly As we are interested in singular RIFs φ =p/p, we assume there exists at least one ζ ∈ T d such that p(ζ) = 0. A result of Pascoe [18,Corollary 1.7] shows that if we assume p is irreducible, then any zero p on T d gives rise to a singularity of φ. We restrict attention to the class of such p for which we have the additional property that Z(p) ∩ T d is finite; we call the corresponding φ =p/p finite-singularity RIFs.
where each L p loc (T d ) is a standard local Lebesgue space on the d-torus. The global z d -derivative integrability index of φ is the maximum of all the local z d -derivative integrability indices of the finite-singularity RIF φ.
Because of the argument principle, ∂φ ∂z d is integrable for any RIF so the assumption that p ≥ 1 is justified; see [7,5] for details. In a similar way, we can define z j -derivative indices. To keep this note as elementary as possible, we focus on the z d -derivative integrability index of a (n, 1) RIF.
It is not a straight-forward task to determine local z j -derivative indices of a d-variable RIF, or their global counterparts. Two-dimensional RIFs are much better understood than their general d-dimensional counterparts: for instance, the z 1 and z 2 -derivative indices of a RIF coincide when d = 2, but this is false when d ≥ 3, and their values are determined by a geometric characteristic of p at its zeros. See [7] and [10] for comprehensive presentations of the two-variable theory.
As explained in [9], the z d -derivative integrability of a polydegree (n, 1) RIF φ is controlled by the rate at which the zero set ofp approaches T d from inside D d . To make this statement precise, we return to the one-variable function φζ and note that the L p norm of the derivative of a Möbius transformation is proportional to the distance to T of the point ψ 0 ∈ D for which φζ(ψ 0 ) = 0; see [8,Lemma 4.2].
Therefore, we set Note that since φ was assumed to be a finite-singularity RIF, the polynomialp 1 has no zeros in T d−1 ; otherwise Z(p) ∩ T d would contain a vertical line [9, Section 3], which is impossible since zeros ofp on T d are also zeros of p. Hence the vanishing of ρ φ near a singularity is determined by the vanishing of its numerator. As a consequence of this discussion and [9, Theorem 2.1], we obtain the following criterion.
Polydegree (n, 1) rational inner functions and 2 × 2 matrices. Suppose φ =p/p is a finite-singularity RIF of polydegree (n, 1), and consider, forζ ∈ T d−1 fixed, the one-variable function Then, φζ(z d ) is a rational function in D, which attains unimodular boundary values at every point ζ d ∈ T by a theorem of Knese [16,Theorem C]. Hence φζ is either a Möbius transformation of the unit disk, or else φζ(z d ) is constant, and equal to some element of T. The former obtains generically, but the latter possibility certainly occurs on some exceptional sets, as can be checked by considering φ d (1, . . . , 1, ζ d ), where φ d is the function in (1.2). Guided by this discussion, we make the following definition.
The slice determinant of φ is the function P φ : T d−1 → C given by Formally, the numerator and the denominator of φζ(z d ) can be read off from M φ (ζ)(z d , 1) T . The slice determinant allows us to detect singularities of φ as well as their finer properties. The third assertion essentially amounts to a computation. Namely, Observing that ζ j = 1/ζ j , j = 1, . . . , d−1, and examining the definition of reflection polynomials, the expression on the right-hand side can be rewritten (in standard multi-index notation) aŝ The result now follows after taking moduli and appealing to Theorem 1.

Compositions and local properties of singularities
Given an (n, 1) finite-singularity RIF, we define the following sequence of functions. See [21] for a fuller study of dynamical properties of mappings, especially skew-products, whose components are RIFs.
Definition 3. Let φ =p/p be a finite-singularity RIF of polydegree (n, 1). Then φ 2 : D d → C is defined as The functions φ N are clearly rational and holomorphic in D d . As can be seen from (2.1) and (2.2), the z j -degree of φ N is at most N · n j for j = 1, . . . , d−1, and deg z d (φ N ) ≤ 1. One complication that may arise is that the numerator and the denominator of the composite function may initially share a common factor. We always assume any such factors present are cancelled, in which case we get a polydegree drop in φ N . Lemma 3. Suppose φ N =p N /p N is as in Definition 3 and does not experience a polydegree drop. Then φ N is a finite-singularity RIF with the same singularities as φ.
Proof. Since φ maps T d onto T, Knese's theorem implies that each (φ N ) * is unimodular. Hence φ N is inner.
Next, forζ ∈ T d−1 , computing the slice matrix of φ N ζ amounts to taking the matrix power M N φ (ζ) = M φ (ζ) · · · M φ (ζ), see [12]. The assumption that φ N has full polydegree implies there are no common factors in the matrices that would be cancelled in φ N . Then, by multiplicativity of determinants, det M N φ (ζ) vanishes if and only if det M φ (ζ) does. Thus, theζ-coordinates of the singularities of φ N are the same as those of φ. Since φ N has degree 1 in z d , and since φ has a singularity on the line {ζ} × T, each suchζ determines a unique η ∈ T such that (ζ, η) ∈ T d is a singularity of φ N .
The following example illustrates that common factors can be eliminated by rotating φ by a suitable factor e ia , a ∈ R, or in other words by replacingp by e iap . Doing this only affects P φ up to a unimodular factor.
If N ∈ N and φ N has full polydegree, then the RIF φ N =p N /p N has local z d -derivative integrability index equal to 1 + q * /N near 1.
Proof. By Lemma 3, φ N is a RIF with the same singularities as φ, and in particular φ N has a singularity at 1. Sincep N and p N have no common factors that can be cancelled, the slice matrix of φ N is equal to M N φ , the N-fold power of the slice matrix of φ. Hence the order of vanishing of the slice determinant of φ N is equal to N times the order of vanishing of the slice determinant of φ. In other words, ∂φ is finite for U ⊃ 1 sufficiently small. By our assumption on φ, this holds if N(1 − p) > −q * and fails when N(1 − p) < −q * , and the result follows.

Applications
Finding extraneous zeros of two-variable RIF denominators.
In [19], Pascoe presents a way of constructing two-variable RIFs with at least one singularity where the local contact order can be prescribed to take any value 2n, n ∈ N. (Strictly speaking, the construction is given in the setting of the bi-upper half-plane, but it can readily be transferred to the bidisk by means of conjugation by a suitable Möbius map. See [8,Section 7].) In particular, any positive even integer is the contact order of some RIF in D 2 . However, Pascoe's construction may produce additional singularities in φ and, to the author's knowledge, does not appear give any immediate information about their location or nature. In principle, this can be addressed by finding all zeros of the two-variable denominator p, and then using the techniques in [8,10] to determine the associated contact orders. By examining the matrix-valued function ζ 1 → M φ (ζ 1 ) we can detect any such extraneous singularities and determine their contact orders in a fairly simple way. First, we compute P φ (ζ 1 ) = det M φ (ζ 1 ) and find the zeros {ζ 1 1 , . . . ζ s 1 } of the one-variable polynomial P φ that are located on the unit circle. Plugging these values into p, we find the point ζ 2 ∈ T at which the polynomial p(ζ j 1 , z 2 ) vanishes as a function of z 2 . Finally, the order of vanishing of P φ gives us the z 2 -contact order of φ at each singularity. By [8,Section 4], this is equal to the z 1 -contact order of φ as well, allowing us to read off the derivative integrability of φ at each singularity.
Example 7 (Example 7.4 of [8]). Consider the two-variable RIF which is obtained using Pascoe's method. His construction guarantees that φ has a singularity at (−1, −1) with contact order equal to 4. The slice matrix associated with φ is and has determinant We immediately discern that φ has an additional singularity at (1, −1), with contact order equal to 2, as was checked in an ad hoc way in [8].