On the half-plane property and the Tutte group of a matroid

TL;DR

This paper explores the half-plane property of matroids using stable polynomials, introduces a method involving the Tutte group to relate the weak and strong properties, and classifies certain matroids based on these properties.

Contribution

It provides a systematic method to reduce the weak half-plane property to the half-plane property for large classes of matroids using the Tutte group.

Findings

No projective geometry has the WHPP.

A binary matroid has the WHPP if and only if it is regular.

T_8 and R_9 do not have the WHPP.

Abstract

A multivariate polynomial is stable if it is non-vanishing whenever all variables have positive imaginary parts. A matroid has the weak half-plane property (WHPP) if there exists a stable polynomial with support equal to the set of bases of the matroid. If the polynomial can be chosen with all of its nonzero coefficients equal to one then the matroid has the half-plane property (HPP). We describe a systematic method that allows us to reduce the WHPP to the HPP for large families of matroids. This method makes use of the Tutte group of a matroid. We prove that no projective geometry has the WHPP and that a binary matroid has the WHPP if and only if it is regular. We also prove that T_8 and R_9 fail to have the WHPP.