  Let S be an abelian semigroup, written additively. Let A be a finite subset
of S. We denote the cardinality of A by |A|. For any positive integer h, the
sumset hA is the set of all sums of h not necessarily distinct elements of A.
We define 0A = {0}. If A_1,...,A_r, and B are finite sumsets of A and
h_1,...,h_r are nonnegative integers, the sumset h_1A + ... + h_rA_r + B is the
set of all elements of S that can be represented in the form u_1 + ... + u_r +
b, where u_i \in h_iA_i and b \in B. The growth function of this sumset is
\gamma(h_1,...,h_r) = |h_1A + ... + h_rA_r + B|. Applying the Hilbert function
for graded modules over graded algebras, where the grading is over the
semigroup of r-tuples of nonnegative integers, we prove that there is a
polynomial p(t_1,...,t_r) such that \gamma(h_1,...,h_r) = p(t_1,...,t_r) if
min(h_1,...,h_r) is sufficienlty large.

Number Theory;Commutative Algebra;Algebraic Geometry