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- Title
Periodicity of Non-Central Integral Arrangements Modulo Positive Integers.
- Authors
Kamiya, Hidehiko; Takemura, Akimichi; Terao, Hiroaki
- 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 ()] 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 ()] to illustrate our results.
- Subjects
INTEGRALS; LINEAR algebra; MATRICES (Mathematics); PARTIALLY ordered sets; POLYNOMIALS; LATTICE theory; MODULES (Algebra); DIVISOR theory
- Publication
Annals of Combinatorics, 2011, Vol 15, Issue 3, p449
- ISSN
0218-0006
- Publication type
Article
- DOI
10.1007/s00026-011-0105-6