Gruber P. Convex and Discrete Geometry

312 Convex Polytopes
Embed E d into E d+1 as usual (first d coordinates) and let q1,...,qd+1 ∈ Z d+1
be the vertices of the lattice simplex T = S + (0,...,0, 1) in the hyperplane
{x : xd+1 = 1}. The lattice Z d+1 has determinant d(Z d+1 ) = 1. Let L be the
sublattice of Z d+1 with basis {q1,...,qd+1}.ThesetF ={α1q1 +···+αd+1qd+1 :
0 ≤ αi < 1} is a fundamental parallelotope of L. Thus, for the determinant of L
we have d(L) = Vd+1(F), where Vd+1 stands for the volume in E d+1 . Then the
following hold.
(2) For each u ∈ Z d+1 , there is precisely one point v ∈ F ∩ Z d+1 such that
u ∈ v + L. Further, Vd+1(F) = #(F ∩ Z d+1 ) = h, say.
If u = β1q1 +···+βd+1qd+1 ∈ Z d+1 , then v = (β1 −⌊β1⌋)q1 +···+(βd+1 −
⌊βd+1⌋)qd+1 is the unique point v ∈ F ∩ Z d+1 with u ∈ v + L. The index of the
sublattice (subgroup) L in the lattice (group) Z d+1 is thus #(F ∩ Z d+1 ). Since the
index is also equal to d(L)/d(Z d+1 ) = Vd+1(F), we see that Vd+1(F) = #(F ∩
Z d+1 ), concluding the proof of (2). Compare Sect. 21.3. For each u ∈ nT ∩ Z d+1 ,
according to (2), we have
�d+1
(3) u = v + miqi with suitable integers mi ≥ 0.
i=1
To see this, note that u = β1q1 +···+βd+1qd+1 with βi ≥ 0 and put mi =⌊βi⌋.
Considering the last coordinate, (3) shows that
�d+1
(4) n = vd+1 + mi with integers mi ≥ 0.
i=1
Conversely, given v ∈ F ∩ Z d+1 , any solution of (4) in integers mi ≥ 0 gives rise to
a unique point u ∈ nT ∩ Z d+1 with u ∈ v + L. The number of such solutions of (4)
equals the coefficient of t n in the power series
t vd+1 2 d+1 t
(1 + t + t +···) =
vd+1
(1 − t) d+1
= t vd+1
� � � � �
−d − 1 −d − 1
1 + (−t) + (−t)
1
2
2 �
+···
= t vd+1
� � � � �
d + 1 d + 2
1 + t + t
1
2
2 �
+··· ,
where we have applied Newton’s binomial series. Hence
(5) L(nS)(= L(nT)) equals the coefficient of tn in the power series
� � � � �
d + 1 d + 2
q(t) 1 + t + t
1
2
2 �
+··· , where q(t) = �
v∈F∩Z d+1
t vd+1 .
To determine the degree of q(·), note that each v ∈ F ∩ Z d+1 can be represented
in the form v = α1q1 +···αd+1qd+1 where 0 ≤ αi < 1. Each qi has last coordinate