A Combinatorial Reciprocity Theorem for Hyperplane Arrangements
A Combinatorial Reciprocity Theorem for Hyperplane Arrangements
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DOI：
10.4153/cmb-2010-004-7
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发表时间：
2006-10
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影响因子：
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通讯作者：
Christos A. Athanasiadis
中科院分区：
文献类型：
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作者：
Christos A. Athanasiadis
Abstract Given a nonnegative integer $m$ and a finite collection $A$ of linear forms on ${{\mathbb{Q}}^{d}}$ , the arrangement of affine hyperplanes in ${{\mathbb{Q}}^{d}}$ defined by the equations $\alpha \left( x \right)\,=\,k$ for $\alpha \,\in \,A$ and integers $k\,\in \,\left[ -m,\,m \right]$ is denoted by ${{A}^{m}}$ . It is proved that the coefficients of the characteristic polynomial of ${{A}^{m}}$ are quasi-polynomials in $m$ and that they satisfy a simple combinatorial reciprocity law.
Abstract Given a nonnegative integer $m$ and a finite collection $A$ of linear forms on ${{\mathbb{Q}}^{d}}$ , the arrangement of affine hyperplanes in ${{\mathbb{Q}}^{d}}$ defined by the equations $\alpha \left( x \right)\,=\,k$ for $\alpha \,\in \,A$ and integers $k\,\in \,\left[ -m,\,m \right]$ is denoted by ${{A}^{m}}$ . It is proved that the coefficients of the characteristic polynomial of ${{A}^{m}}$ are quasi-polynomials in $m$ and that they satisfy a simple combinatorial reciprocity law.