Polynomials with no zeros on the bidisk

Knese, Greg

doi:10.2140/apde.2010.3.109

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Polynomials with no zeros on the bidisk

Greg Knese

Vol. 3 (2010), No. 2, 109–149

DOI: 10.2140/apde.2010.3.109

Abstract

We prove a detailed sums of squares formula for two-variable polynomials with no zeros on the bidisk $D^{2}$ , extending previous such formulas by Cole and Wermer and by Geronimo and Woerdeman. Our 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 Fejér–Riesz factorizations, analytic extension theorems for a class of bordered curves called distinguished varieties, and Pick interpolation on the bidisk.

Keywords

bidisk, Christoffel–Darboux, sums of squares, Fejér–Riesz, orthogonal polynomials, distinguished varieties, Pick interpolation, Andô's inequality, Bernstein–Szegő measures, torus, stable polynomials

Mathematical Subject Classification 2000

Primary: 42C05

Secondary: 47A57, 46C07, 42B05, 14M12

Milestones

Received: 23 October 2008

Revised: 20 October 2009

Accepted: 3 December 2009

Published: 8 June 2010

Authors

Greg Knese
University of California, Irvine Department of Mathematics Irvine CA 92697-3875 United States
http://www.math.uci.edu/~gknese

Vol. 3, No. 2, 2010