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arXiv · 0809.3087

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

Julius Borcea·Petter Brändén
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Abstract

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 Pólya-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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Julius Borcea, Petter Brändén. 2008-09-18. The Lee-Yang and Pólya-Schur Programs. II. Theory of Stable Polynomials and Applications. https://doi.org/10.1002/cpa.20295

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