Newton polyhedron, Hilbert polynomial, and sums of finite sets
type: Item
instance of: scholarly article
title: Newton polyhedron, Hilbert polynomial, and sums of finite sets
zbMATH Open document ID: 0809.13012
DOI: 10.1007/BF01077066
publication date: 1994-05-18T00:00:00Z
review text: 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 \(B\).
reviewed by: Mirela Ştefănescu
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 14M25
zbMATH DE Number: 560995
zbMATH Keywords: number of elements
zbMATH Keywords: Hilbert polynomials
zbMATH Keywords: reduced Newton polyhedron
author: A. G. Khovanskii
MaRDI profile type: Publication
cites work: Remarks on the perturbation theory for problems of Mathieu type
cites work: Three coupled oscillators: Mode-locking, global bifurcations and toroidal chaos
cites work: Resonance regions for Mathieu type dynamical systems on a torus
cites work: Resonance regions for families of torus maps
full work available at URL: https://doi.org/10.1007/bf01077066
OpenAlex ID: W1976765258
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 560995