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

TL;DR

This paper classifies all matroids with up to 8 elements possessing the half-plane property and explores related classes, revealing their structural properties and closure under minors and faces of Newton polytopes.

Contribution

It provides a complete classification of small matroids with the half-plane property and establishes closure properties for related classes motivated by semidefinite programming.

Findings

All matroids with up to 8 elements with the half-plane property are classified.

Some 9-element matroids have or lack the half-plane property.

Certain classes of matroids and polynomials are closed under minors and faces of Newton polytopes.

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.