Annals of Combinatorics

, 15:449 | Cite as

Periodicity of Non-Central Integral Arrangements Modulo Positive Integers

  • Hidehiko KamiyaEmail author
  • Akimichi Takemura
  • Hiroaki Terao


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.

Mathematics Subject Classification

32S22 52C35 


characteristic quasi-polynomial elementary divisor hyperplane arrangement intersection poset 


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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Hidehiko Kamiya
    • 1
    Email author
  • Akimichi Takemura
    • 2
  • Hiroaki Terao
    • 3
  1. 1.Graduate School of EconomicsNagoya UniversityNagoyaJapan
  2. 2.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyUniversity of TokyoTokyoJapan
  3. 3.Department of MathematicsHokkaido UniversitySapporoJapan

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