Local theory of stable polynomials and bounded rational functions of several variables

Abstract

We provide detailed local descriptions of stable polynomials in terms of their homogeneous decompositions, Puiseux expansions, and transfer function realizations. We use this theory to first prove that bounded rational functions on the polydisk possess nontangential limits at every boundary point. We relate higher non-tangential regularity and distinguished boundary behavior of bounded rational functions to geometric properties of the zero sets of stable polynomials via our local descriptions. For a fixed stable polynomial p, we analyze the ideal of numerators q such that q/p is bounded on the bi-upper half plane. We completely characterize this ideal in several geometrically interesting situations including smooth points, double points, and ordinary multiple points of p. Finally, we analyze integrability properties of bounded rational functions and their derivatives on the bidisk.

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

“…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.…”

“…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.…”

“…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 and p(ζ) = 0 vanish without having common factors that cancel out.…”

“…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 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.…”