Gruber P. Convex and Discrete Geometry

Let φn,µn be as in the proof of Proposition 19.3. Then
19 Lattice Polytopes 331
L(nP) = φn(P) = µn(P + H d ) = µn(a0S0 +···+ad Sd + H d )
= a0µn(S0 + H d ) +···+adµn(Sd + H d )
� � � �
n + 0
n + d
= a0φn(S0) +···+adφn(Sd) = a0 +···+ad
0
d
= b0 + b1n +···+bdn d for P ∈ P Z d , n ∈ N,
by Proposition 19.3, where a0,...,ad and thus b0,...,bd are suitable integers,
depending only on P. Thus we may write,
(14) L(nP) = L0(P) + L1(P)n +···+Ld(P)n d for P ∈ P Z d , n ∈ N.
It is easy to check that the mappings Li : P Z d → R are integer unimodular invariant
valuations: let P, Q ∈ P Z d be such that P ∪ Q, P ∩ Q ∈ P Z d . Then L � n(P ∪ Q) � +
L � n(P ∩ Q) � = L(nP) + L(nQ). Represent each of these four expressions as a
polynomial in n according to (14) and compare coefficients. By (14),
Fixing n ∈ N, this shows that
L(mnP) = L0(nP) + L1(nP)m +···+Ld(nP)m d
= L0(P) + L1(P)mn +···+Ld(P)m d n d
for P ∈ P Z d , m, n ∈ N.
Li(nP) = n i Li(P) for P ∈ P Z d , n ∈ N, i = 0,...,d.
Since L(nSi) = ( n+i
i ) is a polynomial in n of degree i, we obtain
Li(Si) �= 0, Li+1(Si) =···= Ld(Si) = 0fori = 0,...,d.
This readily implies that L0,...,Ld are linearly independent. The proof of (13) is
complete.
The theorem finally follows from (12) and (13). ⊓⊔
Ehrhart’s Lattice Point Enumerators Theorem
As simple corollaries of the theorem of Betke and Kneser we get again the
theorem 19.1 of Ehrhart [292, 293] on lattice point enumerators, but without the
information that the constant term in the polynomial is 1.
Corollary 19.1. Let P ∈ P Z d p . Then the following hold:
(i) L(nP) = pP(n) for n ∈ N, where pP is a polynomial of degree d with leading
coefficient V (P).
(ii) L o (nP) = (−1) d pP(−n) for n ∈ N.
Proof. (i) By the Betke–Kneser theorem,