Ehrhart polynomials of matroid polytopes and polymatroids

TL;DR

This paper studies Ehrhart polynomials of matroid and polymatroid polytopes, proving polynomial-time computability for fixed rank and exploring conjectures about their h^*-vectors and coefficients.

Contribution

It establishes polynomial-time algorithms for Ehrhart polynomial computation of fixed-rank matroid polytopes and provides evidence for conjectures on their combinatorial properties.

Findings

Ehrhart polynomials are computable in polynomial time for fixed rank.

The paper offers theoretical and computational support for conjectures on h^*-vectors.

New analysis of Todd polynomials enhances understanding of Ehrhart polynomial evaluation.

Abstract

We investigate properties of Ehrhart polynomials for matroid polytopes, independence matroid polytopes, and polymatroids. In the first half of the paper we prove that for fixed rank their Ehrhart polynomials are computable in polynomial time. The proof relies on the geometry of these polytopes as well as a new refined analysis of the evaluation of Todd polynomials. In the second half we discuss two conjectures about the h^*-vector and the coefficients of Ehrhart polynomials of matroid polytopes; we provide theoretical and computational evidence for their validity.