Period Collapse in Characteristic Quasi-Polynomials of Hyperplane Arrangements
Period Collapse in Characteristic Quasi-Polynomials of Hyperplane Arrangements
复制标题
超平面排列特征拟多项式的周期塌缩
DOI：
10.1093/imrn/rnac104
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发表时间：
2022
影响因子：
1
通讯作者：
Yoshinaga Masahiko
中科院分区：
文献类型：
--
作者：
Higashitani Akihiro;Tran Tan Nhat;Yoshinaga Masahiko
给定一个积分超平面排列，Kamiya-Takemura-Terao [，]引入了特征拟多项式的概念，它列举了以正整数为模的排列的补数的基数。最流行的候选周期的特征准多项式是lcm周期。本文从Ehrhart拟多项式理论中的周期坍缩概念出发，研究了特征拟多项式的周期坍缩问题。当最小周期严格小于lcm周期时，我们说周期坍缩发生在特征拟多项式中。我们的第一个主要结果是，在非中心情况下，关于周期坍缩，任何事情都是可能的：周期坍缩发生在任何维度，发生在任何lcm周期的值上，并且当它不是lcm周期时，最小周期可以是lcm周期的任何真因子。我们的第二个主要结果指出，在中心的情况下，然而，没有周期崩溃是可能的，在任何维度，也就是说，lcm周期总是最小的周期。
Given an integral hyperplane arrangement, Kamiya–Takemura–Terao [, ] 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, occurs for any valueof 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.