Stable polynomials and admissible numerators in product domains

TL;DR

This paper characterizes the ideal of polynomials that produce bounded rational functions near boundary zeros of stable polynomials in product domains, providing explicit descriptions under smoothness and non-degeneracy conditions.

Contribution

It offers a complete description of the ideal of admissible numerators for stable polynomials with smooth zeros and introduces integral closure methods for isolated boundary zeros.

Findings

Explicit ideal descriptions under smoothness conditions

Integral closure characterization for isolated zeros

Constructed multivariate stable polynomials demonstrating sharpness

Abstract

Given a polynomial $p$ with no zeros in the polydisk, or equivalently the poly-upper half-plane, we study the problem of determining the ideal of polynomials $q$ with the property that the rational function $q/p$ is bounded near a boundary zero of $p$. We give a complete description of this ideal of numerators in the case where the zero set of $p$ is smooth and satisfies a non-degeneracy condition. We also give a description of the ideal in terms of an integral closure when $p$ has an isolated zero on the distinguished boundary. Constructions of multivariate stable polynomials are presented to illustrate sharpness of our results and necessity of our assumptions.