The motivic zeta functions of a matroid

Abstract

We introduce motivic zeta functions for matroids. These zeta functions are defined as sums over the lattice points of Bergman fans, and in the realizable case, they coincide with the motivic Igusa zeta functions of hyperplane arrangements. We show that these motivic zeta functions satisfy a functional equation arising from matroid Poincaré duality in the sense of Adiprasito-Huh-Katz. In the process, we obtain a formula for the Hilbert series of the cohomology ring of a matroid, in the sense of Feichtner-Yuzvinsky. We then show that our motivic zeta functions specialize to the topological zeta functions for matroids introduced by van der Veer, and we compute the first two coefficients in the Taylor expansion of these topological zeta functions, providing affirmative answers to two questions posed by van der Veer.

arXiv comment28 pages, 2 figures

References in corpus (7)

• Chow rings of toric varieties defined by atomic lattices
• Matroid polytopes, nested sets and Bergman fans
• Monodromy Filtrations and the Topology of Tropical Varieties
• Grothendieck classes and Chern classes of hyperplane arrangements
• Chern-Schwartz-MacPherson cycles of matroids
• Almost all matroids are non-representable
• Combinatorial analogs of topological zeta functions

Cited by in corpus (3)

• Valuative invariants for large classes of matroids
• Hilbert-Poincaré series of matroid Chow rings and intersection cohomology
• On the geometry of flag Hilbert-Poincaré series for matroids