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

TL;DR

This paper develops a local theory of stable polynomials and bounded rational functions in several variables, linking their boundary behavior and zero set geometry to function regularity and integrability properties.

Contribution

It introduces a detailed local description of stable polynomials and applies this to analyze boundary limits, regularity, and the structure of bounded rational functions in multiple variables.

Findings

Bounded rational functions on the polydisk have non-tangential boundary limits.

The ideal of numerators for bounded quotients is characterized in various geometric cases.

The paper links boundary regularity of functions to the geometry of polynomial zero sets.

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 non-tangential 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.