Ehrhart polynomials of rank two matroids

TL;DR

This paper investigates Ehrhart polynomials of rank two matroids, proving positivity of coefficients, bounding them by minimal and uniform matroids, and establishing real-rootedness and unimodality of certain $h^*$-polynomials.

Contribution

It proves positivity for Ehrhart polynomials of rank two matroids, bounds their coefficients, and confirms real-rootedness and unimodality of $h^*$-polynomials for sparse paving matroids of rank two.

Findings

Ehrhart polynomials of rank two matroids have positive coefficients.

Coefficients of these polynomials are bounded by those of minimal and uniform matroids.

The $h^*$-polynomials of sparse paving matroids of rank two are real-rooted, unimodal, and log-concave.

Abstract

Over a decade ago De Loera, Haws and K\"oppe conjectured that Ehrhart polynomials of matroid polytopes have only positive coefficients and that the coefficients of the corresponding $h^*$-polynomials form a unimodal sequence. The first of these intensively studied conjectures has recently been disproved by the first author who gave counterexamples in all ranks greater or equal to three. In this article we complete the picture by showing that Ehrhart polynomials of matroids of lower rank have indeed only positive coefficients. Moreover, we show that they are coefficient-wise bounded by the Ehrhart polynomials of minimal and uniform matroids. We furthermore address the second conjecture by proving that $h^*$-polynomials of matroid polytopes of sparse paving matroids of rank two are real-rooted and therefore have log-concave and unimodal coefficients. In particular, this shows that the $h^*$-polynomial of the second hypersimplex is real-rooted thereby strengthening a result of De Loera, Haws and K\"oppe.