CHARACTERISTIC-POLYNOMIALS OF SUBSPACE ARRANGEMENTS AND FINITE-FIELDS

Let A be any subspace arrangement in R(n) defined over the integers an d let F-q denote the finite field with q elements. Let q be a large pr ime. We prove that the characteristic polynomial chi(A, q) of A counts the number of points in F-q(n) that do not lie in any of the subspace s of A, viewed as subsets of F-q(n). This observation, which generaliz es a theorem of Blass and Sagan about subarrangements of the B-n, arra ngement, reduces the computation of chi(A, q) to a counting problem an d provides an explanation for the wealth of combinatorial results disc overed in the theory of hyperplane arrangements in recent years. The b asic idea has its origins in the work of Crapo and Rota (1970). We fin d new classes of hyperplane arrangements whose characteristic polynomi als have simple form and very often Factor completely over the nonnega tive integers. (C) 1996 Academic Press, Inc.