Combinatorial analogs of topological zeta functions

TL;DR

This paper introduces a new combinatorial invariant for matroids, called Z(L,s), which generalizes the topological zeta function of polynomials and depends only on the matroid's lattice structure.

Contribution

It defines a novel rational function invariant for matroids that mimics the topological zeta function, independent of resolution choices, linking combinatorics and topology.

Findings

Z(L,s) recovers the topological zeta function for hyperplane arrangements

Z(L,s) is independent of resolution choices

Provides potential tests for matroid realizability

Abstract

In this article we introduce a new matroid invariant, a combinatorial analog of the topological zeta function of a polynomial. More specifically we associate to any ranked, atomic meet-semilattice L a rational function Z(L,s), in such a way that when L is the lattice of flats of a complex hyperplane arrangement we recover the usual topological zeta function. The definition is in terms of a choice of a combinatorial analog of resolution of singularities, and the main result is that Z(L,s) does not depend on this choice and depends only on L. Known properties of the topological zeta function provide a source of potential complex realisability test for matroids.