The Characteristic Quasi-Polynomials of Hyperplane Arrangements over Residually Finite Dedekind Domains

TL;DR

This paper extends the theory of characteristic quasi-polynomials from integer arrangements to arrangements over Dedekind domains, providing algebraic generalizations of known properties and periods.

Contribution

It introduces algebraic generalizations of characteristic quasi-polynomials for arrangements over Dedekind domains, expanding the scope of previous integer-based results.

Findings

Established the period of the characteristic quasi-polynomial over Dedekind domains.

Connected the constituents of the quasi-polynomial to characteristic polynomials of arrangements.

Proved the LCM-period as the minimal period in this algebraic setting.

Abstract

Kamiya, Takemura, and Terao initiated the theory of the characteristic quasi-polynomial of an integral arrangement, which is a function counting the elements in the complement of the arrangement modulo positive integers. They gave a period of the characteristic quasi-polynomial, called the LCM-period, and showed that the first constituent of the characteristic quasi-polynomial coincides with the characteristic polynomial of the corresponding hyperplane arrangement. Recently, Liu, Tran, and Yoshinaga showed that the last constituent of the characteristic quasi-polynomial coincides with the characteristic polynomial of the corresponding toric arrangement. In addition, by using the theory of toric arrangements, Higashitani, Tran, and Yoshinaga proved that the LCM-period is the minimum period of the characteristic quasi-polynomial. In this paper, we study an arrangements over a Dedekind domain such that every residue ring with a nonzero ideal is finite and give algebraic generalizations of the above results.