Stable symmetric polynomials and the Schur-Agler class

TL;DR

This paper characterizes when symmetric, multi-affine, and stable polynomials serve as denominators of rational inner functions within the Schur-Agler class, advancing understanding of multivariable polynomial stability and function theory.

Contribution

It provides a necessary and sufficient condition for symmetric, multi-affine, stable polynomials to be Agler denominators, and refines existing results in three-variable cases.

Findings

Established a criterion for Agler denominators among symmetric, multi-affine, stable polynomials.

Extended Kummert's results to three-variable cases, sharpening previous theorems.

Enhanced understanding of the structure of rational inner functions in the Schur-Agler class.

Abstract

We call a multivariable polynomial an Agler denominator if it is the denominator of a rational inner function in the Schur-Agler class, an important subclass of the bounded analytic functions on the polydisk. We give a necessary and sufficient condition for a multi-affine, symmetric, and stable polynomial to be an Agler denominator and prove several consequences. We also sharpen a result due to Kummert related to three variable, multi-affine, stable polynomials.