Vol. 3, No. 2, 2010

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ISSN: 1948-206X (e-only)
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Polynomials with no zeros on the bidisk

Greg Knese

Vol. 3 (2010), No. 2, 109–149
Abstract

We prove a detailed sums of squares formula for two-variable polynomials with no zeros on the bidisk D2, 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