Ask zeta functions of central hyperplane arrangements

Alec Schmutz

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arXiv · 2606.21393

Ask zeta functions of central hyperplane arrangements

Alec Schmutz

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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-Poincar\'e series of $\mathcal{A}$, obtained as a rank specialisation of the flag Hilbert-Poincar\'e 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-Poincar\'e 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.

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Alec Schmutz. 2026-06-19. Ask zeta functions of central hyperplane arrangements. https://arxiv.org/abs/2606.21393

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