Ask zeta functions of central hyperplane arrangements

Abstract

Given a central hyperplane arrangement $\mathcal{A}$ defined over a field of characteristic zero, we construct matrices of linear forms whose local ask zeta functions are recovered by the Igusa local zeta function of the cone over $\mathcal{A}$. Our construction extends a previously established connection between ask zeta function of hypergraphs and the Igusa local zeta function of Boolean arrangements. From a combinatorial standpoint, we introduce the truncated flag Hilbert-Poincaré series of $\mathcal{A}$, obtained as a rank specialisation of the flag Hilbert-Poincaré series of the cone over $\mathcal{A}$. Whenever $\mathcal{A}$ admits good reduction over the finite field $\mathbb{F}_q$, suitable substitutions of the variables of its truncated flag Hilbert-Poincaré series recover the local ask zeta functions associated with $\mathcal{A}$. Such formulae provide a means to study the analytic properties of the ask zeta functions considered, as well as to derive their reduced and topological relatives.