The motivic zeta functions of a matroid

TL;DR

This paper introduces motivic zeta functions for matroids, establishing their properties, relations to existing invariants, and computing initial coefficients, thus connecting combinatorics, geometry, and topology.

Contribution

It defines motivic zeta functions for matroids, proves their functional equations, and relates them to topological zeta functions, providing new insights and answers to open questions.

Findings

Motivic zeta functions satisfy a functional equation from matroid Poincaré duality.

They coincide with motivic Igusa zeta functions in realizable cases.

First two coefficients of the Taylor expansion of topological zeta functions are computed.

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 Poincar\'{e} 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.