On the Ehrhart Polynomial of Minimal Matroids

Abstract

We provide a formula for the Ehrhart polynomial of the connected matroid of size $n$ and rank $k$ with the least number of bases, also known as a minimal matroid. We prove that their polytopes are Ehrhart positive and $h^{\ast }$-real-rooted (and hence unimodal). We prove that the operation of circuit-hyperplane relaxation relates minimal matroids and matroid polytopes subdivisions, and also preserves Ehrhart positivity. We state two conjectures: that indeed all matroids are $h^{\ast }$-real-rooted, and that the coefficients of the Ehrhart polynomial of a connected matroid of fixed rank and cardinality are bounded by those of the corresponding minimal matroid and the corresponding uniform matroid.

15 pages, 2 figures. To appear in Discrete and Computational Geometry

References in corpus (2)

• Matroid polytopes, nested sets and Bergman fans
• Hypersimplices are Ehrhart Positive

Cited by in corpus (8)

• Matroids are not Ehrhart positive
• Valuative invariants for large classes of matroids
• Hypersimplices are Ehrhart Positive
• Stressed hyperplanes and Kazhdan-Lusztig gamma-positivity for matroids
• Ehrhart polynomials of rank two matroids
• Matroid relaxations and Kazhdan-Lusztig non-degeneracy
• The Merino--Welsh conjecture for split matroids
• Face enumeration for split matroid polytopes