Gruber P. Convex and Discrete Geometry

334 Convex Polytopes
have to be singletons too and thus are vertices of N p and Nq, respectively, say v,w.
Then u = v +w and this is the only possible representation of u as the sum of a point
of N p and a point of Nq. The product of the corresponding monomials bx v1
1 ···xvd d
of p and cx w1
1 ···x wd
d of q is thus a monomial of the form axu1 1 ···xud d and no other
product of a monomial of p and a monomial of q has this form. Hence ax u1
1 ···xud d
is the monomial in pq corresponding to u,oru ∈ N pq. Thus
N p + Nq ⊆ N pq,
concluding the proof. ⊓⊔
A real polynomial on E d is irreducible if it cannot be represented as a product
of two real polynomials on E d , each consisting of more than one monomial. Call a
convex lattice polytope in E d integer irreducible if it cannot be represented as the
sum of two convex lattice polytopes, each consisting of more than one point. As
an immediate consequence of the earlier Proposition 19.4, we obtain the following
irreducibility criterion, see Gao [355].
Theorem 19.7. Let p : E d → R be a real polynomial with Newton polytope N p. If
N p is integer irreducible, then p is irreducible.
Remark. Actually, this result holds for any polynomial p in d variables over any
field F. In addition, irreducibility of p may be sharpened to absolute irreducibility,
i.e. irreducibility in the algebraic closure of F. When N p is not integer irreducible,
then p may still be irreducible. An example is provided by the irreducible real
polynomial p(x, y) = 1 + y + xy + x 2 + y 2 , whose Newton polytope N p =
conv � o,(0, 2), (2, 0) � is integer reducible:
conv{o,(0, 2), (2, 0)} =conv{o,(0, 1), (1, 0)}+conv{o,(0, 1), (1, 0)}.
Gao and Lauder [356] study the integer irreducibility of convex lattice polytopes.
It turns out that the problem to decide whether a convex lattice polytope is integer
irreducible is NP-complete.
The Stepanov–Schmidt Irreducibility Criterion
Clearly, a convex lattice polygon in E 2 , which has an edge of the form [(0, m), (n, 0)]
with m and n relatively prime and which is contained in the triangle conv{o,(0, m),
(n, 0)}, is integer irreducible. This leads to the following criterion of Stepanov and
Schmidt, see W. Schmidt [895], p. 92, and Gao [355].
Corollary 19.2. Let p be a real polynomial in two variables, such that its Newton
polygon N p contains an edge of the form [(0, m), (n, 0)], where m and n are
relatively prime, and is contained in the triangle conv � o,(0, m), (n, 0) � . Then p
is irreducible.
For example, the polynomials x 2 + y 3 and x 3 + y 7 + xy+ xy 2 + x 2 y 2 are irreducible.