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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