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

TL;DR

This paper develops a comprehensive theory of multivariate stable polynomials based on classification theorems, providing a unified framework for Lee-Yang problems across statistical mechanics, combinatorics, and geometric function theory.

Contribution

It extends the Pólya-Schur classification to multivariate polynomials and applies this to solve problems in various mathematical and physical contexts.

Findings

Characterized linear operators preserving non-vanishing properties in multivariate polynomials.

Developed a theory of multivariate stable polynomials.

Provided solutions to Lee-Yang type problems and answered a question on multivariate apolarity.

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 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.