Matroids on Eight Elements with the Half-plane Property and Related Concepts

Abstract

We classify all matroids with at most 8 elements that have the half-plane property, and we provide a list of some matroids on 9 elements that have, and that do not have the half-plane property. Furthermore, we prove that several classes of matroids and polynomials that are motivated by the theory of semidefinite programming are closed under taking minors and under passing to faces of the Newton polytope.