Characteristic quasi-polynomials of truncated arrangements

Abstract

Given an (affine) integral arrangement $\mathcal{A}$ in $\mathbb{R}^n$, the reduction of $\mathcal{A}$ modulo an arbitrary positive integer $q$ naturally yields an arrangement $\mathcal{A}_q$ in $\mathbb{Z}_q^n$. Our primary objective is to study the combinatorial aspects of the restriction $\mathcal{A}^{(B,\bm b)}$ to the solution space of $B\bm x=\bm b$, and its reduction $\mathcal{A}_q^{(B,\bm b)}$ modulo $q$. This work generalizes the earlier results of Kamiya, Takemura and Terao, as well as Chen and Wang. The purpose of this paper is threefold as follows. Firstly, we derive an explicit counting formula for the cardinality of the complement $M\big(\mathcal{A}_q^{(B,\bm b)}\big)$ of $\mathcal{A}_q^{(B,\bm b)}$; and prove that for all positive integers $q>q_0$, this cardinality coincides with a quasi-polynomial $\chi^{\text{quasi}}\big(\mathcal{A}^{(B,\bm b)},q\big)$ in $q$ with a period $\rho_C$. Secondly, we weaken Chen and Wang's original hypothesis $a \mid b$ to a strictly more general condition $\gcd(a,\rho_C)\mid \gcd(b,\rho_C)$, and introduce the concept of combinatorial equivalence for positive integers. Within this framework, we establish three unified comparison relations: between the unsigned coefficients of $\chi^{\text{quasi}}\big(\mathcal{A}^{(B,\bm b)},a\big)$ and $\chi^{\text{quasi}}\big(\mathcal{A}^{(B,\bm b)},b\big)$; between the unsigned coefficients of distinct constituents of $\chi^{\text{quasi}}\big(\mathcal{A}^{(B,\bm b)},q\big)$; and between the cardinalities of $M\big(\mathcal{A}_q^{(B,\bm b)}\big)$ and $M\big(\mathcal{A}_{pq}^{(B,\bm b)}\big)$. Thirdly, using our method, we revisit the enumerative aspects of group colorings and nowhere-zero nonhomogeneous form flows from the early work of Forge, Zaslavsky and Kochol.