Sums of finite sets, orbits of commutative semigroups, and Hilbert functions
type: Item
instance of: scholarly article
title: Sums of finite sets, orbits of commutative semigroups, and Hilbert functions
zbMATH Open document ID: 0855.13011
DOI: 10.1007/BF01080008
publication date: 1997-02-03T00:00:00Z
review text: This paper takes as a starting point Hilbert's theorem that the dimensions of the \(n\)-th filtered subspace of a graded module over a commutative polynomial algebra is given by a polynomial in \(n\), for \(n\) large enough. This is seen to be a special case of results about the size of finite subsets \(B + nA\) of a commutative semigroup \(G\), where \(B\) and \(A\) are fixed and \(n\) varies; and this in turn is a special case of the central theorem of the paper, which considers the sum of the values of a multiplicative character \(\chi : G \to \mathbb{C}\) over a subset \(B + nA\), and shows it to be given by an expression of the form \(\sum q^n_i P_i (n)\) for large \(n\). Here, the \(q_i\) are the values which \(\chi\) takes on \(A\), and \(P_i\) is a polynomial of degree less than the number of points of \(A\) at which \(\chi\) takes the value \(q_i\). On the way to proving this the author studies interesting ideas concerning the orbits of the semigroup \((\mathbb{N}, +)^{(t)}\) acting on a set, and makes crucial use of the classical fact that this semigroup is Noetherian. The present paper is a sequel to a paper by the author [Funct. Anal. Appl. 26, No. 4, 276-281 (1992); translation from Funkts. Anal. Prilozh. 26, No. 4, 57-63 (1992; Zbl 0809.13012)].
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 859609
zbMATH Keywords: Hilbert-Samuel polynomial
zbMATH Keywords: orbits of commutative semigroups
zbMATH Keywords: filtered subspace of a graded module
zbMATH Keywords: polynomial algebra
author: A. G. Khovanskii
reviewed by: Kenneth A. Brown
MaRDI profile type: Publication
cites work: Quasimodes generated by characters of a dynamical group, and deformation of the Kirillov form
cites work: Points entiers dans les polyèdres convexes
cites work: Q4031179
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published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 859609