A Combinatorial Reciprocity Theorem for Hyperplane Arrangements

Given a nonnegative integer m and a finite collection. A of linear forms on Q(d), the arrangement of affine hyperplanes in Q(d) defined by the equations alpha(x) = k for alpha is an element of A and integers k is an element of [-m, m] is denoted by A(m). It is proved that the coefficients of the characteristic polynomial of A(m) are quasi-polynomials in m and that they satisfy a simple combinatorial reciprocity law.