The Lee-Yang and Pólya-Schur Programs. II. Theory of Stable Polynomials and Applications

Details The Lee-Yang and Pólya-Schur Programs. II. Theory of Stable Polynomials and Applications The Lee-Yang and Pólya-Schur Programs. II. Theory of Stable Polynomials and Applications

2008-09-18

arxiv.org/abs/0809.3087v1

Comm. Pure Appl. Math. 62 (2009), no. 12, 1595-1631

Mathematics

Complex Variables

32 pages

Scientific paper

10.1002/cpa.20295

In the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing in products of open circular domains. For such sets this completes the multivariate generalization of the classification program initiated by P\'olya-Schur for univariate real polynomials. We build on these classification theorems to develop here a theory of multivariate stable polynomials. Applications and examples show that this theory provides a natural framework for dealing in a uniform way with Lee-Yang type problems in statistical mechanics, combinatorics, and geometric function theory in one or several variables. In particular, we answer a question of Hinkkanen on multivariate apolarity.

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