On the Ehrhart Polynomial of Minimal Matroids

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.