%0 Journal Article
%T The Lee-Yang and P¨®lya-Schur Programs. II. Theory of Stable Polynomials and Applications
%A Julius Borcea
%A Petter Br£¿nd¨¦n
%J Mathematics
%D 2008
%I arXiv
%R 10.1002/cpa.20295
%X 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.
%U http://arxiv.org/abs/0809.3087v1