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Séminaire Lotharingien de Combinatoire, 86B.27 (2022), 12 pp.

# Joshua Maglione and Christopher Voll

# Flag Hilbert-Poincaré Series and Igusa
Zeta Functions of Hyperplane Arrangements

**Abstract.**
We introduce and study a class of multivariate rational functions
associated with hyperplane arrangements, called flag
Hilbert-Poincaré series. These series are intimately connected
with Igusa local zeta functions of products of linear polynomials,
and their motivic and topological relatives. Our main results
include a self-reciprocity result for central arrangements defined
over fields of characteristic zero. We also prove combinatorial
formulae for a specialization of the flag Hilbert-Poincaré series
for irreducible Coxeter arrangements of types *A*,
*B*, and *D* in terms of total partitions of the
respective types. We show that a different specialization of the
flag Hilbert-Poincaré series, which we call the coarse flag
Hilbert-Poincaré series, exhibits intriguing nonnegativity
features and - in the case of Coxeter arrangements - connections
with Eulerian polynomials. For numerous classes and examples of
hyperplane arrangements, we determine their (coarse) flag
Hilbert-Poincaré series. Some computations were aided by a
\SageMath~package we developed.

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

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