A combinatorial reciprocity theorem for hyperplane arrangements

Abstract

Given a nonnegative integer m and a finite collection A of linear forms on Qd, the arrangement of affine hyperplanes in Qd defined by the equations a(x) = k for α € A and integers k € [-m, m] is denoted by Am. It is proved that the coefficients of the characteristic polynomial of Am are quasi-polynomials in m and that they satisfy a simple combinatorial reciprocity law. © 2010 Canadian Mathematical Society.