Topological Zeta Functions of Matroids: Operations and Computations

TL;DR

This paper studies the topological zeta functions of matroids, revealing their behavior under matroid operations, and introduces a new invariant called girth to describe their Taylor coefficients.

Contribution

It provides recurrence relations for zeta functions under matroid operations and characterizes Taylor coefficients using the newly defined girth invariant.

Findings

Recurrence relation for M"obius inversion of matroids.

Explicit formulas for zeta functions of truncations and free extensions.

Girth invariant characterizes Taylor coefficients of zeta functions.

Abstract

The topological zeta function of a matroid is a rational function as well as a valuative invariant of the matroid, encoding rich combinatorial information. We analyze topological zeta functions of matroids from the vantage point of several matroid operations and operations on lattices of flats. We prove a clean recurrence relation for the M\"obius inversion and use it to describe the topological zeta function of the truncation and free extension of a matroid in relation with that of the original matroid. We also characterize the Taylor coefficients of the topological zeta functions for matroids in terms of a matroid invariant, which we call the girth, and generalize an earlier result by Jensen-Kutler-Usatine.