Characteristic quasi-polynomials of truncated arrangements
Ying Cao, Houshan Fu
TL;DR
This work extends the theory of characteristic quasi-polynomials to non-central, truncated integral arrangements, providing an explicit counting formula for the complement cardinality over finite rings and proving that it is a quasi-polynomial in the modulus with period $\rho_C$. It introduces a gcd-based combinatorial equivalence framework and proves three unified comparison results: between the coefficients of the quasi-polynomial constituents, between the constituents themselves, and between the counting functions for different moduli. By weakening previous divisibility assumptions to $\gcd(a,\rho_C)\mid\gcd(b,\rho_C)$ and employing combinatorial equivalence, the paper unifies and extends prior results of Chen–Wang and Kamiya–Takemura–Terao, and reinterprets classical graph-coloring and flow problems in terms of truncated arrangements. The results have broad applications to modular colorings and nowhere-zero flows, connecting to Athanasiadis’ finite-field method and Zaslavsky’s affinographic framework, and offering new tools for enumerative combinatorics on lattices, toric arrangements, and related combinatorial geometries.
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 $χ^{\text{quasi}}\big(\mathcal{A}^{(B,\bm b)},q\big)$ in $q$ with a period $ρ_C$. Secondly, we weaken Chen and Wang's original hypothesis $a \mid b$ to a strictly more general condition $\gcd(a,ρ_C)\mid \gcd(b,ρ_C)$, and introduce the concept of combinatorial equivalence for positive integers. Within this framework, we establish three unified comparison relations: between the unsigned coefficients of $χ^{\text{quasi}}\big(\mathcal{A}^{(B,\bm b)},a\big)$ and $χ^{\text{quasi}}\big(\mathcal{A}^{(B,\bm b)},b\big)$; between the unsigned coefficients of distinct constituents of $χ^{\text{quasi}}\big(\mathcal{A}^{(B,\bm b)},q\big)$; 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.