Periodicity of Non-Central Integral Arrangements Modulo Positive Integers |
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Authors: | Hidehiko Kamiya Akimichi Takemura Hiroaki Terao |
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Institution: | 1. Graduate School of Economics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan 2. Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan 3. Department of Mathematics, Hokkaido University, North 10, West 8, Kita-ku, Sapporo, 060-0810, Japan
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Abstract: | An integral coefficient matrix determines an integral arrangement of hyperplanes in
\mathbbRm{\mathbb{R}^m} . After modulo q reduction ${(q \in {\mathbb{Z}_{ >0 }})}${(q \in {\mathbb{Z}_{ >0 }})} , the same matrix determines an arrangement Aq{\mathcal{A}_q} of “hyperplanes” in
\mathbbZmq{\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 Aq{\mathcal{A}_q} in
\mathbbZmq{\mathbb{Z}^m_q} is a quasi-polynomial in ${q \in {\mathbb{Z}_{ >0 }}}${q \in {\mathbb{Z}_{ >0 }}} . Moreover, they proved in the central case that the intersection lattice of Aq{\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
^(B)]m0,a]{\hat{\mathcal{B}}_m^{0,a]}} of Athanasiadis J. Algebraic Combin. 10(3), 207–225 (1999)] to illustrate our results. |
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