An identity for the coefficients of characteristic polynomials of hyperplane arrangements

2020/08/15 by Kabluchko, Zakhar · 1 citation
#51F15 #51M04 #51M20. Secondary: 52A55 #52A22 #52B11 #60D05 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Primary: 52C35 #Probability (math.PR)

paper · doi:10.48550/arxiv.2008.06719

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∈ \mathbb R^d and a chamber P the metric projection of x onto P is the unique point y∈ 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 dim(x,P). We prove that for every given k∈ \0,…, d\, the number of chambers P for which 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].

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• Interpreting the (signed) chromatic polynomial coefficients via hyperplane arrangements

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