Gruber P. Convex and Discrete Geometry

Sufficient Conditions for Polynomiality
19 Lattice Polytopes 321
We begin with two criteria for polynomiality. Both seem to be known. A result similar
to the first criterion is due to Carroll [191]. I am grateful to Iskander Aliev [22]
for this reference. Here N ={0, 1,...}.
Lemma 19.1. Let k ∈ N and p : N d → R such that for any d − 1 variables fixed, p
is (the restriction of) a real polynomial of degree at most k in the remaining variable.
Then p is a real polynomial in all d variables of degree at most k.
Proof. We prove the lemma in case d = 2. The same idea, together with a simple
induction argument, yields the general case.
By assumption,
k�
(1) p(m, n) = ai(m)n i for m, n ∈ N,
i=0
where the ai(m) are suitable coefficients. Thus, in particular,
p(m, j) =
k�
ai(m) j i for m ∈ N, j = 0,...,k.
i=0
Cramer’s rule and the Gram determinant then imply that
k�
(2) ai(m) = bij p(m, j) for m ∈ N, i = 0,...,k,
j=0
with suitable real coefficients bij. Thus
p(m, n) =
k�
bij p(m, j)n i for m, n ∈ N,
i, j=0
by (1) and (2). Since, by the assumption of the lemma, p(m, j) is a polynomial in
m (∈ N) for each j = 0, 1,...,k, p(m, n) is also a polynomial in m, n (∈ N). ⊓⊔
Lemma 19.2. Let q : N → R be such that q(n + 1) − q(n) is (the restriction of) a
real polynomial for n ∈ N. Then q is a real polynomial in one variable.
Proof. Let
q(n + 1) − q(n) = a0 + a1n +···+akn k for n ∈ N,
with suitable coefficients ai. Then
q(n + 1) − q(1) = q(n + 1) − q(n) + q(n) − q(n − 1) +···+q(2) − q(1)
= a0 + a1n +···+akn k + a0 + a1(n − 1) +···+ak(n − 1) k +···
= a0(1 + 1 +···+1) + a1(1 + 2 +···+n) +···+ak(1 + 2 k +···+n k ).