论文标题
超平面布置特征多项式系数的身份
An identity for the coefficients of characteristic polynomials of hyperplane arrangements
论文作者
论文摘要
考虑$ \ Mathbb r^d $中的Affine Explanes的有限收集。超级平面$ \ mathbb r^d $将$ \ mathbb r^d $剖析到有限的许多多面体室中。对于\ mathbb r^d $和abscor $ p $的点$ x \，$ x $在$ p $上的公制投影是p $ y \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y \ y的，将欧几里得距离最小化至$ x $。公制投影包含在$ p $的独特面部的相对内部中，其尺寸用$ \ text {dim}（x，p）$表示。我们证明，对于每个给定的$ k \ in \ {0，\ ldots，d \} $，$ \ text {dim}（x，p）= k $的钱伯斯$ p $的数量不取决于$ x $的选择，除了某些lebesgue null设置。此外，该数字等于超平面布置特征多项式的$ k $ Th系数的绝对值。在反射布置的特殊情况下，这证明了Drton和Klivans的猜想[对反射布置的特征多项式的几何解释，Proc。阿米尔。数学。 Soc。，138（8）：2873-2887，2010]。
Consider a finite collection of affine hyperplanes in $\mathbb R^d$. The hyperplanes dissect $\mathbb R^d$ into finitely many polyhedral chambers. For a point $x\in \mathbb R^d$ and a chamber $P$ the metric projection of $x$ onto $P$ is the unique point $y\in P$ minimizing the Euclidean distance to $x$. The metric projection is contained in the relative interior of a uniquely defined face of $P$ whose dimension is denoted by $\text{dim}(x,P)$. We prove that for every given $k\in \{0,\ldots, d\}$, the number of chambers $P$ for which $\text{dim}(x,P) = k$ does not depend on the choice of $x$, with an exception of some Lebesgue null set. Moreover, this number is equal to the absolute value of the $k$-th coefficient of the characteristic polynomial of the hyperplane arrangement. In a special case of reflection arrangements, this proves a conjecture of Drton and Klivans [A geometric interpretation of the characteristic polynomial of reflection arrangements, Proc. Amer. Math. Soc., 138(8): 2873-2887, 2010].
