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 $\alpha (x)=k$ for $\alpha \in \mathcal{A}$ 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.

7 pages