Periodicity of non-central integral arrangements modulo positive integers

TL;DR

This paper extends known periodicity and counting results from central hyperplane arrangements to non-central arrangements modulo positive integers, providing new insights into their combinatorial structure.

Contribution

It generalizes the periodicity and quasi-polynomial cardinality results from central to non-central arrangements, broadening the understanding of hyperplane arrangements modulo q.

Findings

Cardinality of arrangements modulo q is a quasi-polynomial in non-central cases.

The intersection lattice of arrangements exhibits periodicity beyond the central case.

Application to specific arrangements illustrates the theoretical results.

Abstract

An integral coefficient matrix determines an integral arrangement of hyperplanes in R^m. After modulo q reduction, the same matrix determines an arrangement A_q of "hyperplanes" in Z^m. In the special case of central arrangements, Kamiya, Takemura and Terao [J. Algebraic Combin., to appear] showed that the cardinality of the complement of A_q in Z_q^m is a quasi-polynomial in q. Moreover, they proved in the central case that the intersection lattice of 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_m^{[0,a]} of Athanasiadis [J. Algebraic Combin. Vol.10 (1999), 207-225] to illustrate our results.