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Periodicity of hyperplane arrangements with integral coefficients modulo positive integers

Hidehiko Kamiya·Akimichi Takemura·Hiroaki Terao
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Abstract

We study central hyperplane arrangements with integral coefficients modulo positive integers $q$. We prove that the cardinality of the complement of the hyperplanes is a quasi-polynomial in two ways, first via the theory of elementary divisors and then via the theory of the Ehrhart quasi-polynomials. This result is useful for determining the characteristic polynomial of the corresponding real arrangement. With the former approach, we also prove that intersection lattices modulo $q$ are periodic except for a finite number of $q$'s.

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Hidehiko Kamiya, Akimichi Takemura, Hiroaki Terao. 2007-04-02. Periodicity of hyperplane arrangements with integral coefficients modulo positive integers. https://arxiv.org/abs/math/0703904

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