Combinatorial analogs of topological zeta functions

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.