 
Summary: Canad. Math. Bull. Vol. 53 (1), 2010 pp. 310
doi:10.4153/CMB20100047
c Canadian Mathematical Society 2010
A Combinatorial Reciprocity Theorem
for Hyperplane Arrangements
Christos A. Athanasiadis
Abstract. Given a nonnegative integer m and a finite collection A of linear forms on Qd, the ar
rangement of affine hyperplanes in Qd defined by the equations (x) = k for A and integers
k [m, m] is denoted by Am. It is proved that the coefficients of the characteristic polynomial of
Am are quasipolynomials in m and that they satisfy a simple combinatorial reciprocity law.
1 Introduction
Let V be a ddimensional vector space over the field Q of rational numbers and A
be a finite collection of linear forms on V which spans the dual vector space V
.
We denote by Am
the essential arrangement of affine hyperplanes in V defined by
the equations (x) = k for A and integers k [m, m] (we refer to [9, 13]
for background on hyperplane arrangements). Thus A0
consists of the linear hyper
planes which are the kernels of the forms in A and Am
