Characteristic Polynomials of Subspace Arrangements and Finite Fields
Characteristic Polynomials of Subspace Arrangements and Finite Fields
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DOI：
10.1006/aima.1996.0059
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发表时间：
1996-09
影响因子：
1.7
通讯作者：
Christos A. Athanasiadis
中科院分区：
文献类型：
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作者：
Christos A. Athanasiadis
设A是在整数上定义的任意子空间排列，设fq表示有qelement的有限域。取一个大质数。我们证明了A的特征多项式χ(A, q)计算了Fnq中不位于作为Fnq子集的A的任何子空间中的点的个数。这一观察推广了Blass和Sagan关于b排列的子排列的一个定理，将χ(a, q)的计算简化为一个计数问题，并为近年来超平面排列理论中发现的大量组合结果提供了解释。其基本思想起源于Crapo和Rota（1970）的工作。我们发现了一类新的超平面排列，它们的特征多项式具有简单的形式，并且常常完全因子于非负整数上。
Let A be any subspace arrangement in Rndefined over the integers and let Fqdenote the finite field withqelements. Letqbe a large prime. We prove that the characteristic polynomialχ(A, q) of A counts the number of points in Fnqthat do not lie in any of the subspaces ofA, viewed as subsets of Fnq. This observation, which generalizes a theorem of Blass and Sagan about subarrangements of the Bnarrangement, reduces the computation ofχ(A, q) to a counting problem and provides an explanation for the wealth of combinatorial results discovered in the theory of hyperplane arrangements in recent years. The basic idea has its origins in the work of Crapo and Rota (1970). We find new classes of hyperplane arrangements whose characteristic polynomials have simple form and very often factor completely over the nonnegative integers.