Polynomials with no zeros on the bidisk

Polynomials with no zeros on the bidisk

Abstract: We prove a detailed sums of squares formula for two variable polynomials with no zeros on the bidisk 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.

Polynomials defining distinguished varieties

Polynomials defining distinguished varieties.  (submitted for publication)

Abstract: Using a sums of squares formula for two variable
polynomials with no zeros on the bidisk, we are able to give a new
proof of a representation formula for distinguished varieties.  For
distinguished varieties with no singularities on the two-torus, we
are able to provide extra details about the representation formula
and use this to prove a bounded extension theorem.

Bernstein-Szegő measures on the two dimensional torus

Bernstein-Szeg\”o measures on the two dimensional torus, Indiana Univ. Math. J. 57 No. 3 (2008), 1353–1376

Function theory on the Neil parabola

The paper “Function theory on the Neil parabola” has appeared in print. Here is a link to the project euclid page for it:

Function theory on the Neil parabola

A Schwarz lemma on the polydisk

My paper “A Schwarz lemma on the polydisk” has “officially” appeared on the Proceedings of the AMS website.

A Schwarz lemma on the polydisk