Period Collapse in Characteristic Quasi-Polynomials of Hyperplane Arrangements

Abstract

Abstract Given an integral hyperplane arrangement, Kamiya–Takemura–Terao [11, 12] introduced the notion of characteristic quasi-polynomial, which enumerates the cardinality of the complement of the arrangement modulo a positive integer. The most popular candidate for periods of the characteristic quasi-polynomials is the lcm period. In this paper, we initiate a study of period collapse in characteristic quasi-polynomials stemming from the concept of period collapse in the theory of Ehrhart quasi-polynomials. We say that period collapse occurs in a characteristic quasi-polynomial when the minimum period is strictly less than the lcm period. Our first main result is that in the non-central case, with regard to period collapse anything is possible: period collapse occurs in any dimension $\ge 1$, occurs for any value $\ge 2$ of the lcm period, and the minimum period when it is not the lcm period can be any proper divisor of the lcm period. Our second main result states that in the central case, however, no period collapse is possible in any dimension, that is, the lcm period is always the minimum period.