Lee-Yang Problems and The Geometry of Multivariate Polynomials

TL;DR

This paper classifies linear operators on multivariate polynomials that preserve non-vanishing properties in open circular domains, extending classical univariate results and aiding Lee-Yang problems across various mathematical fields.

Contribution

It completes the multivariate generalization of the Pólya-Schur classification program and offers a unified framework for Lee-Yang type problems in multiple disciplines.

Findings

Classified all linear operators preserving non-vanishing in open circular domains.

Provides a framework for Lee-Yang problems in statistical mechanics, combinatorics, and geometric function theory.

Extends univariate polynomial results to multivariate cases.

Abstract

We describe all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing in open circular domains. This completes the multivariate generalization of the classification program initiated by P\'olya-Schur for univariate real polynomials and provides a natural framework for dealing in a uniform way with Lee-Yang type problems in statistical mechanics, combinatorics, and geometric function theory. This is an announcement with some of the main results in arXiv:0809.0401 and arXiv:0809.3087.