Newton polyhedron, Hilbert polynomial, and sums of finite sets
If \(G\) is a commutative semigroup, one may define, for two subsets \(A\) and \(B\) of \(G\), the sum \(A+B\) and the sum of \(N\) copies of \(A\) denoted by \(N*A\). In the paper, it is shown that if \(A\) and \(B\) are finite sets, then the number of elements of the set \(B + N* A\) with \(N\) large enough is a polynomial in \(N\) whose degree is less than the number of elements of \(A\) (theorem 1). The polynomials in this result and in its generalization (theorem 2) are Hilbert polynomials of certain graded modules over the ring of polynomials in several indeterminates. When \(G\) is an abelian group without elements of finite order, denoting by \(G(A)\) the subgroup generated by the differences of the elements of \(A\), \(A \subseteq G\), then \(G(A)\) is isomorphic to \(\mathbb{Z}^ n\) where \(n\) is the rank of \(G(A)\) and \(\overline A = A - a\) is included into \(G(A)\) for all \(a \in A\). The reduced Newton polyhedron of \(A \subseteq G\) is the convex hull, in the space \(\mathbb{R}^ n\) containing the lattice \(\mathbb{Z}^ n\), of the image of \(A\) under an isomorphism of the group \(G(A)\) on \(\mathbb{Z}^ n\). The main result on the reduced Newton polyhedron is the following: The number of points in \(B+N*A\) divided by \(N^ n\) tends as \(N \to \infty\), to the product of the volume of the reduced Newton polyhedron of \(A\) and the number \(i(A,B)\) of cosets of the group \(G\) modulo the subgroup \(G(A)\) which contain points of the set \(B\).
scientific article; zbMATH DE number 560995