On the geometry of flag Hilbert-Poincar\'e series for matroids

TL;DR

This paper extends the concept of flag Hilbert-Poincaré series to matroids, analyzing their properties through geometric and combinatorial methods, and establishes bounds related to Eulerian polynomials and topes.

Contribution

It introduces a new class of series for matroids, explores their properties in oriented cases, and provides criteria for non-orientability based on these series.

Findings

Numerators are bounded below by Eulerian polynomials.

Equality occurs if and only if all topes are simplicial.

Provides a criterion for non-orientability of matroids.

Abstract

We extend the definition of coarse flag Hilbert--Poincar\'e series to matroids; these series arise in the context of local Igusa zeta functions associated to hyperplane arrangements. We study these series in the case of oriented matroids by applying geometric and combinatorial tools related to their topes. In this case, we prove that the numerators of these series are coefficient-wise bounded below by the Eulerian polynomial and equality holds if and only if all topes are simplicial. Moreover this yields a sufficient criterion for non-orientability of matroids of arbitrary rank.