Multivariate stable polynomials: theory and applications

Abstract

Univariate polynomials with only real roots -- while special -- do occur often enough that their properties can lead to interesting conclusions in diverse areas. Due mainly to the recent work of two young mathematicians, Julius Borcea and Petter Br\"and\'en, a very successful multivariate generalization of this method has been developed. The first part of this paper surveys some of the main results of this theory of "multivariate stable" polynomials -- the most central of these results is the characterization of linear transformations preserving stability of polynomials. The second part presents various applications of this theory in complex analysis, matrix theory, probability and statistical mechanics, and combinatorics.