Gruber P. Convex and Discrete Geometry

322 Convex Polytopes
Now, take into account that the sum of the ith-order arithmetic series 1+2 i +···+n i
is a polynomial in n of degree i + 1, where the coefficients depend on the Bernoulli
numbers. Hence q(n +1)−q(1) and thus q(n +1) and then also q(n) is a polynomial
in n of degree k + 1. ⊓⊔
The McMullen–Bernstein Theorem on Lattice Points in Sums of Lattice
Polytopes
is as follows.
Theorem 19.4. Let P1,...,Pm ∈ P Z d p . Then L(n1 P1+···+nm Pm) is a polynomial
in n1,...,nm ∈ N of degree d.
Proof. We begin with some simple preparations. The inclusion–exclusion formula
for finite sets is as follows.
(3) Let A1,...,Am be finite sets. Then
# �
Ai = �
#Ai − �
#(Ai ∩ A j) + �
#(Ai ∩ A j ∩ Ak) −···
i
i
i< j
i< j 0 and that (5) holds in case where dim Q = 0, 1,...,l − 1.
We have to prove it for dim Q = l. Since translations of P and Q by vectors of Z d
do not affect (5), we may assume that
o ∈ P, Q.
u∈U