Ehrhart polynomials of rank two matroids. (English) Zbl 1496.52016

Summary: Over a decade ago De Loera, Haws and Köppe conjectured that Ehrhart polynomials of matroid polytopes have only positive coefficients and that the coefficients of the corresponding \(h^\ast \)-polynomials form a unimodal sequence. The first of these intensively studied conjectures has recently been disproved by the first author who gave counterexamples in all ranks greater than or equal to three. In this article we complete the picture by showing that Ehrhart polynomials of matroids of lower rank have indeed only positive coefficients. Moreover, we show that they are coefficient-wise bounded by the Ehrhart polynomials of minimal and uniform matroids. We furthermore address the second conjecture by proving that \(h^\ast \)-polynomials of matroid polytopes of sparse paving matroids of rank two are real-rooted and therefore have log-concave and unimodal coefficients. In particular, this shows that the \(h^\ast \)-polynomial of the second hypersimplex is real-rooted, thereby strengthening a result of De Loera, Haws and Köppe.


52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
05A15 Exact enumeration problems, generating functions
05B35 Combinatorial aspects of matroids and geometric lattices
26C10 Real polynomials: location of zeros
Full Text: DOI arXiv


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