Gruber P. Convex and Discrete Geometry

332 Convex Polytopes
(15) L(nP) = L0(P) +···+Ld(P)n d = pP(n) for n ∈ N,
where pP is a polynomial of degree ≤ d. We have to show that Ld(P) = V (P).
Using the formula for the calculation of Jordan measure in Sect. 7.2, it follows that
�
1
V (P) = lim # P ∩
n→∞ nd 1
n Zd
�
1
= lim
n→∞ nd #(nP ∩ Zd )
L(nP)
= lim
n→∞ nd = Ld(P).
(ii) We first show that L o is a valuation on P Z d .LetQ, R ∈ P Z d be such that
Q ∪ R, Q ∩ R ∈ P Z d . Considering the cases Q ⊆ R, orR ⊆ Q;dimQ = dim R =
dim(Q ∩ R) + 1; dim Q = dim R = dim(Q ∩ R), it is easy to see that L o (Q ∪
R) + L o (Q ∩ R) = L o (Q) + L o (R), i.e. L o is a valuation. Clearly, L o is integer
unimodular invariant. Thus the theorem of Betke and Kneser implies that
(16) L o (nP) =
d�
ai Li(nP) =
i=0
d�
ain i Li(P) for n ∈ N
i=0
with suitable coefficients ai independent of P and n. In order to determine the coefficients
ai take the lattice cube [0, 1] d instead of P. Then
(17) L � n[0, 1] d� = (n + 1) d d�
= n i
� �
d
for n ∈ N,
i
(18) L o� n[0, 1] d� = (n − 1) d =
i=0
d�
n i (−1) d−i
i=0
� �
d
for n ∈ N.
i
Now compare (15) and (17) and also (16) and (18) to see that ai = (−1) d−i . Thus
L o d�
(nP) = (−1) d−i Li(P)n i = (−1) d
d�
Li(P)(−n) i
i=0
= (−1) d pP(n) for n ∈ N
by (16) and (15), concluding the proof of (ii). ⊓⊔
i=0
19.5 Newton Polytopes: Irreducibility of Polynomials
and the Minding–Kouchnirenko–Bernstein Theorem
Let p = p(x1,...,xd) be a polynomial in d variables over R, C, or some other field.
Its Newton polytope N p is the convex hull of all points (u1,...,ud) ∈ Z d , such that,
in p, there is a monomial of the form
ax u1
1 ···xud d where a �= 0.
The Newton polytope N p conveys important properties of the polynomial p.Newton
polytopes turned out to be of interest in algebra, algebraic geometry and numerical
analysis. They constitute a bridge between convexity and algebraic geometry.