Flag Hilbert-Poincar\'e series of hyperplane arrangements and their Igusa zeta functions

TL;DR

This paper introduces flag Hilbert-Poincaré series linked to hyperplane arrangements, exploring their properties, connections with Igusa zeta functions, and providing combinatorial formulas for Coxeter arrangements, supported by computational tools.

Contribution

It defines and studies flag Hilbert-Poincaré series, establishing reciprocity, combinatorial formulas, and nonnegativity properties, advancing understanding of hyperplane arrangements and their zeta functions.

Findings

Self-reciprocity for central arrangements over characteristic zero fields.

Combinatorial formulas for types A, B, D Coxeter arrangements.

Nonnegativity and connections with Eulerian polynomials for certain series.

Abstract

We introduce and study a class of multivariate rational functions associated with hyperplane arrangements, called flag Hilbert-Poincar\'e 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-Poincar\'e series for irreducible Coxeter arrangements of types $\mathsf{A}$, $\mathsf{B}$, and $\mathsf{D}$ in terms of total partitions of the respective types. We show that a different specialization of the flag Hilbert-Poincar\'e series, which we call the coarse flag Hilbert-Poincar\'e 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-Poincar\'e series. Some computations were aided by a SageMath package we developed.