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.
|Number of pages||29|
|Journal||Journal of the London Mathematical Society|
|State||Published - Mar 2021|
Bibliographical noteFunding Information:
D.J. was supported by NSF DMS‐1601896. J.U. was supported by NSF DMS‐1702428 and an NSF graduate research fellowship.
© 2020 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.
- 05B35 (primary)
- 14T05 (secondary)
ASJC Scopus subject areas
- Mathematics (all)