Period collapse in characteristic quasi-polynomials of hyperplane arrangements

TL;DR

This paper investigates when the characteristic quasi-polynomials of hyperplane arrangements exhibit period collapse, revealing that in non-central cases collapse can occur arbitrarily, while in central cases it cannot.

Contribution

It establishes the conditions under which period collapse occurs or does not occur in characteristic quasi-polynomials of hyperplane arrangements.

Findings

Period collapse occurs in non-central arrangements in any dimension.

In non-central cases, the minimum period can be any proper divisor of the lcm period.

In central arrangements, the lcm period is always the minimum period.

Abstract

Given an integral hyperplane arrangement, Kamiya-Takemura-Terao (2008 & 2011) 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 period 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 of the lcm period $\ge 2$, 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.