The motivic zeta functions of a matroid

David Jensen, Max Kutler, Jeremy Usatine

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

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.

Original languageEnglish
Pages (from-to)604-632
Number of pages29
JournalJournal of the London Mathematical Society
Volume103
Issue number2
DOIs
StatePublished - Mar 2021

Bibliographical note

Publisher Copyright:
© 2020 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.

Keywords

  • 05B35 (primary)
  • 14E18
  • 14T05 (secondary)

ASJC Scopus subject areas

  • General Mathematics

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