Grace-like polynomials

TL;DR

This paper introduces and analyzes Grace-like polynomials, a class of polynomials with specific zero-location properties related to circle separation, connecting to statistical mechanics and graph theory.

Contribution

It defines Grace-like polynomials and provides properties and characterizations, clarifying their role in zero distribution problems.

Findings

Characterization of Grace-like polynomials

Connection to Lee-Yang circle theorem

Properties related to zero locations

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 $P(z_1,...,w_n)\ne0$ whenever there is a circle in ${\bf C}$ separating $z_1,...,z_m$ from $w_1,...,w_n$. A number of properties and characterizations of these polynomials are obtained.