  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.
