A Combinatorial Reciprocity Theorem for Hyperplane Arrangements

http://dx.doi.org/10.4153/CMB-2010-004-7
Canad. Math. Bull. 53(2010), 3-10

  Published:2009-12-04
 Printed: Mar 2010

Given a nonnegative integer $m$ and a finite collection $\mathcal A$ of linear forms on $\mathcal Q^d$, the arrangement of affine hyperplanes in $\mathcal Q^d$ 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.