Computing bases for rings of permutation-invariant polynomials
type: Item
instance of: scholarly article
title: Computing bases for rings of permutation-invariant polynomials
zbMATH Open document ID: 0832.13006
author: Manfred Göbel
published in: Journal of Symbolic Computation
publication date: 1996-02-20T00:00:00Z
review text: Let \(R[X_1, \ldots, X_n]\) be the polynomial ring in \(X_1, \ldots, X_n\) where \(R\) is a unitary commutative ring. Given any subgroup \(G\) of the group of permutations of the set \(\{X_1, \ldots, X_n\}\), call \(S : = R[X_1, \ldots, X_n]^G\) the \(R\)-algebra of \(G\)-invariant polynomials. In this paper it is showed how to obtain a basis \(B\) for the \(R\)-algebra \(S\) and it is given an algorithm which allows to represent any element of \(S\) as a linear combination of elements of \(B\) with coefficients in \(R [\sigma_1, \ldots, \sigma_n]\) (where \(\sigma_1, \ldots, \sigma_n\) are the elementary symmetric polynomials). Moreover it is proved that the elements of \(B\) are polynomials of degree bounded by \(\max \{n, n(n - 1)/2\}\). The construction given is independent of the ground ring \(R\).
reviewed by: Alessandro Logar
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 13P99
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 68Q42
zbMATH DE Number: 792833
zbMATH Keywords: basis for the algebra of invariant polynomials
zbMATH Keywords: algorithm
MaRDI profile type: Publication
full work available at URL: jsco.1995.1017
OpenAlex ID: W2050626380
DOI: 10.1006/JSCO.1995.1017
description: scientific article; zbMATH DE number 792833