An identity for the coefficients of characteristic polynomials of hyperplane arrangements

TL;DR

This paper establishes a geometric interpretation of the coefficients of the characteristic polynomial of hyperplane arrangements, linking them to the count of chambers with a specific face dimension, and confirms a conjecture for reflection arrangements.

Contribution

It proves that the absolute value of the k-th coefficient of the characteristic polynomial equals the number of chambers with a given face dimension, generalizing a conjecture for reflection arrangements.

Findings

Number of chambers with a fixed face dimension is independent of the point chosen.

The k-th coefficient of the characteristic polynomial equals the count of chambers with face dimension k.

Confirms a conjecture for reflection arrangements.

Abstract

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].