Gruber P. Convex and Discrete Geometry

19 Lattice Polytopes 319
The original proof of (ii) by Macdonald was rather complicated. Using Ehrhart’s
reciprocity theorem 19.1(ii), the following proof of (ii) is along the lines of the proof
of Proposition (i) and much shorter.
Proof. (i) By Proposition (i) in Ehrhart’s theorem 19.1,
(4) L(nP) = pP(n) for n ∈ N, where pP is a polynomial of degree d with
leading coefficient V (P) and constant term 1.
Since pP(i) = L(iP) �= 0fori = 0,...,d, Lagrange’s theorem on partial fractions
shows that
d�
pP(t) a j
(5)
=
d�
t − j
(t − i) j=0
i=0
with suitable coefficients a j. Then
Put t = j to see that
Multiplying (5) by
pP(t) =
L( jP) = pP( j) = a j
d�
i=0
i�= j
d�
j=0
a j
d�
(t − i).
i=0
i�= j
( j − i), or a j = (−1)d− j L( jP)
(d − j)! j!
d�
(t −i) and comparing the coefficient of td on both sides shows
i=0
that
�
d
�
d !V (P) = L(dP) − L
1
� (d − 1)P � +−···+(−1) d
by (4).
(ii) Propositions (i) and (ii) in Ehrhart’s theorem 19.1 together imply that
(6) M(nP) = 1 � � �
2 L(nP) + Lo 1
(nP) = 2 pP(n) + (−1) d pP(−n) � = qP(n),
say, for n ∈ N, where qP is a polynomial of degree d with leading coefficient
V (P), second coefficient 0, and constant term 1 2 + 1 2 (−1)d .
Since qP(i) = 1 � �
2 L(iP) + Lo (iP) �= 0fori = 0,...,d − 1, Lagrange’s theorem
yields
qP(t)
=
�
(t − i)
d−1
i=0
with suitable coefficients b j. Then
d−1
�
j=0
b j
+ V (P)
t − j
.