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Canad. Math. Bull. Vol. 53 (1), 2010 pp. 310 doi:10.4153/CMB-2010-004-7
 

Summary: Canad. Math. Bull. Vol. 53 (1), 2010 pp. 310
doi:10.4153/CMB-2010-004-7
c Canadian Mathematical Society 2010
A Combinatorial Reciprocity Theorem
for Hyperplane Arrangements
Christos A. Athanasiadis
Abstract. Given a nonnegative integer m and a finite collection A of linear forms on Qd, the ar-
rangement of affine hyperplanes in Qd defined by the equations (x) = k for A and integers
k [-m, m] is denoted by Am. It is proved that the coefficients of the characteristic polynomial of
Am are quasi-polynomials in m and that they satisfy a simple combinatorial reciprocity law.
1 Introduction
Let V be a d-dimensional vector space over the field Q of rational numbers and A
be a finite collection of linear forms on V which spans the dual vector space V
.
We denote by Am
the essential arrangement of affine hyperplanes in V defined by
the equations (x) = k for A and integers k [-m, m] (we refer to [9, 13]
for background on hyperplane arrangements). Thus A0
consists of the linear hyper-
planes which are the kernels of the forms in A and Am

  

Source: Athanasiadis, Christos - Department of Mathematics, University of Athens

 

Collections: Mathematics