Characteristic polynomials of subspace arrangements and finite fields

Abstract

Let A be any subspace arrangement in ℝn defined over the integers and let Fq denote the finite field with q elements. Let q be a large prime. We prove that the characteristic polynomial X(A , q) of A counts the number of points in Fnq that do not lie in any of the subspaces of A, viewed as subsets of Fnq. This observation, which generalizes a theorem of Blass and Sagan about subarrangements of the script Bn arrangement, reduces the computation of X(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. © 1996 Academic Press, Inc.