Periodicity of Non-Central Integral Arrangements Modulo Positive Integers
| Autor: | Hidehiko Kamiya, Hiroaki Terao, Akimichi Takemura |
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| Rok vydání: | 2011 |
| Předmět: |
Discrete mathematics
Complement (group theory) Reduction (recursion theory) Lattice (group) 32S22 52C35 Combinatorics Matrix (mathematics) Cardinality Hyperplane FOS: Mathematics Mathematics - Combinatorics Discrete Mathematics and Combinatorics Combinatorics (math.CO) Arrangement of hyperplanes Algebraic number Mathematics |
| Zdroj: | Annals of Combinatorics. 15:449-464 |
| ISSN: | 0219-3094 0218-0006 |
| DOI: | 10.1007/s00026-011-0105-6 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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