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A Combinatorial Reciprocity Theorem for Hyperplane Arrangements

Published:2009-12-04
Printed: Mar 2010
• Christos A. Athanasiadis
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Abstract

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.
 MSC Classifications: 52C35 - Arrangements of points, flats, hyperplanes [See also 32S22] 05E99 - None of the above, but in this section

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