On the Ehrhart Polynomial of Minimal Matroids
Luis Ferroni
Abstract
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 $$h^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>h</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -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 $$h^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>h</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -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.