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Séminaire Lotharingien de Combinatoire, 86B.68 (2022), 11 pp.

# Lukas Kühne and Joshua Maglione

# On the Geometry of Flag Hilbert-Poincaré Series for 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.

Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.

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