Grace-like polynomials

Abstract

Results of somewhat mysterious nature are known on the location of zeros of certain polynomials associated with statistical mechanics (Lee-Yang circle theorem) and also with graph counting. In an attempt at clarifying the situation we introduce and discuss here a natural class of polynomials. Let $P(z_1,...,z_m,w_1,...,w_n)$ be separately of degree 1 in each of its $m+n$ arguments. We say that $P$ is a Grace-like polynomial if $P(z_1,...,w_n)\ne 0$ whenever there is a circle in $\mathbf{C}$ separating $z_1,...,z_m$ from $w_1,...,w_n$. A number of properties and characterizations of these polynomials are obtained.

14 pages

Cited by in corpus (5)

• The Lee-Yang and Pólya-Schur Programs. I. Linear Operators Preserving Stability
• The Lee-Yang and Pólya-Schur Programs. II. Theory of Stable Polynomials and Applications
• Anomalous scaling and Lee-Yang zeroes in Self-Organized Criticality
• The early historical roots of Lee-Yang theorem
• A converse to the Grace--Walsh--Szegő theorem