Periodicity of non-central integral arrangements modulo positive integers
Abstract
An integral coefficient matrix determines an integral arrangement of hyperplanes in Rm. After modulo q reduction, the same matrix determines an arrangement Aq of "hyperplanes" in Zm. In the special case of central arrangements, Kamiya, Takemura and Terao [J. Algebraic Combin., to appear] showed that the cardinality of the complement of Aq in Zqm is a quasi-polynomial in q. Moreover, they proved in the central case that the intersection lattice of Aq 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 Bm[0,a] of Athanasiadis [J. Algebraic Combin. Vol.10 (1999), 207-225] to illustrate our results.
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