Periodicity of non-central integral arrangements modulo positive integers

Lattice point counts for the Shi arrangement and other affinographic hyperplane arrangements

Hyperplanes of the form x_j = x_i + c are called affinographic. For an affinographic hyperplane arrangement in R^n, such as the Shi arrangement, we study the function f(M) that counts integral points in [1,M]^n that do not lie in any hyperplane of the arrangement. We show that f(M) is a piecewise polynomial function of positive integers M, composed of terms that appear gradually as M increases. Our approach is to convert the problem to one of counting integral proper colorations of a rooted integral gain graph. An application is to interval coloring in which the interval of available colors fo…

Modular periodicity of the Euler numbers and a sequence by Arnold

For any positive integer $q$, the sequence of the Euler up/down numbers reduced modulo $q$ was proved to be ultimately periodic by Knuth and Buckholtz. Based on computer simulations, we state for each value of $q$ precise conjectures for the minimal period and for the position at which the sequence starts being periodic. When $q$ is a power of $2$, a sequence defined by Arnold appears, and we formulate a conjecture for a simple computation of this sequence.

Syzygies of Unimodular Lawrence Ideals

Infinite hyperplane arrangements whose vertices form a lattice are studied from the point of view of commutative algebra. The quotient of such an arrangement modulo the lattice action represents the minimal free resolution of the associated binomial ideal, which defines a toric subvariety in a product of projective lines. Connections to graphic arrangements and to Beilinson's spectral sequence are explored.

An equivalent formulation of chromatic quasi-polynomials

Given a central integral arrangement, the reduction of the arrangement modulo positive integers $q$ gives rise to a subgroup arrangement in $(\mathbb{Z}/q\mathbb{Z})^\ell$. Kamiya-Takemura-Terao (2008) introduced the notion of characteristic quasi-polynomials, which uses to evaluate the cardinality of the complement of the subgroup arrangement. Chen-Wang (2012) found a similar but more general setting that replacing the integral arrangement by its restriction to a subspace of $\mathbb{R}^\ell$, and evaluating the cardinality of the $q$-reduction complement will also lead to a quasi-polynomial …

The Characteristic Quasi-Polynomials of Hyperplane Arrangements over Residually Finite Dedekind Domains

Kamiya, Takemura, and Terao initiated the theory of the characteristic quasi-polynomial of an integral arrangement, which is a function counting the elements in the complement of the arrangement modulo positive integers. They gave a period of the characteristic quasi-polynomial, called the LCM-period, and showed that the first constituent of the characteristic quasi-polynomial coincides with the characteristic polynomial of the corresponding hyperplane arrangement. Recently, Liu, Tran, and Yoshinaga showed that the last constituent of the characteristic quasi-polynomial coincides with the char…

Period collapse in characteristic quasi-polynomials of hyperplane arrangements

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 pe…

The characteristic quasi-polynomials of the arrangements of root systems and mid-hyperplane arrangements

Let $q$ be a positive integer. In our recent paper, we proved that the cardinality of the complement of an integral arrangement, after the modulo $q$ reduction, is a quasi-polynomial of $q$, which we call the characteristic quasi-polynomial. In this paper, we study general properties of the characteristic quasi-polynomial as well as discuss two important examples: the arrangements of reflecting hyperplanes arising from irreducible root systems and the mid-hyperplane arrangements. In the root system case, we present a beautiful formula for the generating function of the characteristic quasi-pol…

截断排列的特征准多项式

Characteristic quasi-polynomials of truncated arrangements

研究截断排列的特征准多项式，推导基数公式并建立统一的比较关系。

Lie algebras associated to fiber-type arrangements

Given a hyperplane arrangement in a complex vector space of dimension n, there is a natural associated arrangement of codimension k subspaces in a complex vector space of dimension k*n. Topological invariants of the complement of this subspace arrangement are related to those of the complement of the original hyperplane arrangement. In particular, if the hyperplane arrangement is fiber-type, then, apart from grading, the Lie algebra obtained from the descending central series for the fundamental group of the complement of the hyperplane arrangement is isomorphic to the Lie algebra of primitive…

由整数矩阵生成的码的重量枚举式周期性

Periodicity of weight enumerators for codes generated by an integral matrix

本文研究了由整数矩阵生成的码的重量枚举式周期性，通过拟多项式方法计算最小重量并推导出图论拟多项式与重量枚举式之间的转换公式。

A note on Tutte polynomials and Orlik-Solomon algebras

Let A be a (central) arrangement of hyperplanes in a finite dimension complex vector space V. Let M(A) be the dependence matroid determined by A. The Orlik-Solomon algebra OS(M) of a matroid M is the exterior algebra on the points modulo the ideal generated by circuit boundaries. The algebra OS(M) is isomorphic to the cohomology algebra of the complement in V of the union of the hyperplanes of A. The Tutte polynomial T(x,y) of M is a powerful invariant of the matroid M. When M(A) is a rank three matroid and A is the complexification of a real arrangement, we prove that OS(M) determines T(x,y).…