A criterion for the half-plane property

TL;DR

This paper presents a new criterion for determining the stability of multiaffine real polynomials, enabling verification of the half-plane property for certain matroids that were previously resistant to such analysis.

Contribution

It introduces a necessary and sufficient condition for polynomial stability and applies it to verify the half-plane property for specific matroids.

Findings

Established a criterion for stability of multiaffine real polynomials.

Verified the half-plane property for seven challenging matroids.

Provided a tool for analyzing polynomial stability in combinatorial structures.

Abstract

We establish a convenient necessary and sufficient condition for a multiaffine real polynomial to be stable, and use it to verify that the half-plane property holds for seven small matroids that resisted the efforts of Choe, Oxley, Sokal, and Wagner [5].