A combinatorial reciprocity theorem for hyperplane arrangements

Abstract

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