Periodicity of Non-Central Integral Arrangements Modulo Positive Integers

Abstract

An integral coefficient matrix determines an integral arrangement of hyperplanes in $${\mathbb{R}^m}$$ . After modulo q reduction $${(q \in {\mathbb{Z}_{ >0 }})}$$ , the same matrix determines an arrangement $${\mathcal{A}_q}$$ of “hyperplanes” in $${\mathbb{Z}^m_q}$$ . In the special case of central arrangements, Kamiya, Takemura, and Terao [J. Algebraic Combin. 27(3), 317–330 (2008)] showed that the cardinality of the complement of $${\mathcal{A}_q}$$ in $${\mathbb{Z}^m_q}$$ is a quasi-polynomial in $${q \in {\mathbb{Z}_{ >0 }}}$$ . Moreover, they proved in the central case that the intersection lattice of $${\mathcal{A}_q}$$ is periodic from some q on. The present paper generalizes these results to the case of non-central arrangements. The paper also studies the arrangement $${\hat{\mathcal{B}}_m^{[0,a]}}$$ of Athanasiadis [J. Algebraic Combin. 10(3), 207–225 (1999)] to illustrate our results.