 
Summary: advances in mathematics 122, 193 233 (1996)
Characteristic Polynomials of Subspace
Arrangements and Finite Fields*
Christos A. Athanasiadis
Department of Mathematics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139
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 /(A, q) of A counts the number of points in Fn
q
that do not lie in any of the subspaces of A, viewed as subsets of Fn
q . This
observation, which generalizes a theorem of Blass and Sagan about subarrangements
of the Bn arrangement, reduces the computation of /(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.
