The Lee-Yang and P\'olya-Schur Programs. I. Linear Operators Preserving Stability

TL;DR

This paper characterizes all linear operators that preserve the non-vanishing property of multivariate polynomials within specific domains, extending classical univariate polynomial results to higher dimensions.

Contribution

It provides a comprehensive classification of linear operators preserving stability in multivariate polynomials, solving a longstanding problem in complex analysis and related fields.

Findings

Complete characterization of stability-preserving linear operators.

Extension of classical univariate polynomial results to multivariate cases.

Resolution of a long-standing classification problem.

Abstract

In 1952 Lee and Yang proposed the program of analyzing phase transitions in terms of zeros of partition functions. Linear operators preserving non-vanishing properties are essential in this program and various contexts in complex analysis, probability theory, combinatorics, and matrix theory. We characterize all linear operators on finite or infinite-dimensional spaces of multivariate polynomials preserving the property of being non-vanishing whenever the variables are in prescribed open circular domains. In particular, this solves the higher dimensional counterpart of a long-standing classification problem originating from classical works of Hermite, Laguerre, Hurwitz and P\'olya-Schur on univariate polynomials with such properties.