A combinatorial reciprocity theorem for hyperplane arrangements
Abstract
Given a nonnegative integer
m
and a finite collection
A
of linear forms on
Q
d
, the arrangement of affine hyperplanes in
Q
d
defined by the equations
α(x)=k
for
α∈A
and integers
k∈[−m,m]
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.