Polynomials with no zeros on a face of the bidisk

TL;DR

This paper introduces a Hilbert space geometric framework to characterize positive bivariate trigonometric polynomials that can be expressed as squares of stable two-variable polynomials with no zeros on a face of the bidisk, providing new characterizations and sums of squares decompositions.

Contribution

It offers novel Hilbert space and operator theoretic characterizations for stable bivariate polynomials and generalizes existing sums of squares results for such polynomials.

Findings

Characterization via split-shift orthogonality condition

Operator-theoretic sums of squares decomposition

Generalization of Cole-Wermer and Schur-Cohn sums of squares results

Abstract

We present a Hilbert space geometric approach to the problem of characterizing the positive bivariate trigonometric polynomials that can be represented as the square of a two variable polynomial possessing a certain stability requirement, namely no zeros on a face of the bidisk. Two different characterizations are given using a Hilbert space structure naturally associated to the trigonometric polynomial; one is in terms of a certain orthogonal decomposition the Hilbert space must possess called the "split-shift orthogonality condition" and another is an operator theoretic or matrix condition closely related to an earlier characterization due to the first two authors. This approach allows several refinements of the characterization and it also allows us to prove a sums of squares decomposition which at once generalizes the Cole-Wermer sums of squares result for two variable stable polynomials as well as a sums of squares result related to the Schur-Cohn method for counting the roots of a univariate polynomial in the unit disk.