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The Lee-Yang and P\'olya-Schur Programs. II. Theory of Stable Polynomials and Applications
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.Comment: 32 page
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