Polynomials with no zeros on the bidisk

TL;DR

This paper extends sums of squares formulas for two-variable polynomials with no zeros on the bidisk, linking to orthogonal polynomials and applications in factorization, extension, and interpolation.

Contribution

It provides a detailed sums of squares formula for bidisk polynomials, addressing uniqueness and ideal structures, extending prior univariate results.

Findings

Extended sums of squares formula for bidisk polynomials

Applications to Fejér-Riesz factorizations and analytic extension

New insights into Pick interpolation on the bidisk

Abstract

We prove a detailed sums of squares formula for two variable polynomials with no zeros on the bidisk $\mathbb{D}^2$ extending previous versions of such a formula due to Cole-Wermer and Geronimo-Woerdeman. The formula is related to the Christoffel-Darboux formula for orthogonal polynomials on the unit circle, but the extension to two variables involves issues of uniqueness in the formula and the study of ideals of two variable orthogonal polynomials with respect to a positive Borel measure on the torus which may have infinite mass. We present applications to two variable Fej\'er-Riesz factorizations, analytic extension theorems for a class of bordered curves called distinguished varieties, and Pick interpolation on the bidisk.