A combinatorial reciprocity theorem for hyperplane arrangements

Details A combinatorial reciprocity theorem for hyperplane arrangements A combinatorial reciprocity theorem for hyperplane arrangements

2006-10-16

arxiv.org/abs/math/0610482v1

Mathematics

Combinatorics

7 pages

Scientific paper

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.

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