Stable symmetric polynomials and 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.