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 [10]. We prove that their polytopes are Ehrhart positive and h * -real-rooted (and hence unimodal). We use our formula for these Ehrhart polynomials to prove that the operation of circuit-hyperplane relaxation of a matroid preserves Ehrhart positivity. We state two conjectures: that indeed all matroids are h * -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.

Ehrhart polynomials of rank two matroids

“…This result proves a conjecture posed in [7] by the first author for the case of matroids of rank 2. The key to prove Theorem 1.2 is the superadditivity of the polynomials P a i ,n (t) that is provided by Proposition 4.1.…”

“…We observe that Q 2,n agrees with the matroid polytope of the minimal matroid T 2,n . From Proposition 3.2 we therefore obtain an alternative proof for the Ehrhart polynomial of the minimal matroid T 2,n given in [7,Theorem 3…”

“…Castillo and Liu [2] conjectured Ehrhart positivity for the larger class of generalized permutohedra (also known as polymatroids). In [7] the first author conjectured that the coefficients of matroid polytopes are not only positive but are moreover coefficient-wise bounded from below and above by the minimal matroid and the uniform matroid, respectively. Recently, the first author disproved these conjectures simultaneously by providing examples with negative coefficients of matroids whose rank ranges between three and corank three [9].…”

Symmetric decompositions, triangulations and real-rootedness

“…This proves (22). Since h Fr (σ n , x) = p Fr,n,0 (x) = p r,0 n,0 (x), from the recurrence of Proposition 5.6 we get…”

“…is a well studied polynomial of degree at most n with nonnegative coefficients, called the h * -polynomial of P. Stapledon [43] showed that h * (P, x) has a nonnegative symmetric decomposition with respect to n whenever P contains a lattice point in its relative interior (and in fact, to the best of our knowledge, the concept of a symmetric decomposition first appeared in [43]). While the question of unimodality of the h * -polynomial has long been studied [18], its γ-positivity and real-rootedness have been investigated more recently for several special classes of lattice polytopes [12,22,28,29,35,36,38] and the unimodality and real-rootedness of its symmetric decomposition have been addressed too [17,30,37].…”