Solving a system of algebraic equations with symmetries
type: Item
instance of: scholarly article
title: Solving a system of algebraic equations with symmetries
zbMATH Open document ID: 0890.68079
DOI: 10.1016/S0022-4049(97)00011-X
author: Antoine Colin
published in: Journal of Pure and Applied Algebra
publication date: 1997-07-27T00:00:00Z
review text: Let \((F)\) be a system of \(p\) polynomial equations \(F_i({\mathbf X})\in k[{\mathbf X}]\), where \(k\) is a commutative field and \({\mathbf X}:= (X_1,\dots, X_n)\) are indeterminates. Let \(G\) be a subgroup of \(\text{GL}_n(k)\). A polynomial \(P\in k[{\mathbf X}]\) (resp. rational function \(P\in k({\mathbf X})\)) is an invariant of \(G\) if and only if for all \(A\in G\) we have \(A.P= P\). We denote by \(k[{\mathbf X}]^G\) (resp. \(k({\mathbf X})^G\)) the algebra of polynomial (resp. rational function) invariants of \(G\). If \(L\) is another subgroup of \(\text{GL}_n(k)\) such that \(G\subset L\), \(P\) is called a primary invariant of \(G\) relative to \(L\) if and only if \(\text{Stab}_L(P)= G\) (where \(\text{Stab}_L(P)\) is the stabilizer of \(P\) in \(L\)). The paper describes the algebra of the invariants of a finite group and how to express these invariants in terms of a small number of them, from both the Cohen-Macaulay algebra and the field theory points of view. A method is proposed to solve \((F)\) by expressing it in terms of primary invariants \(\Pi_1,\dots, \Pi_n\) (e.g. the elementary symmetric polynomials) and one ``primitive'' secondary invariant. The main thrust of the paper is contained in the following theorem. Let \((F)\) be a set of invariants of \(G\). Let \(L\) be a subgroup of \(\text{GL}_n(k)\) such that \(G\subset L\) and \(k({\mathbf X})^L\) is a purely transcendental extension of \(k\), let \(\Pi_1,\dots, \Pi_n\) be polynomials such that \(k({\mathbf X})^L= k(\Pi_1,\dots, \Pi_n)\), and let \(\Theta\in k[{\mathbf X}]^G\) be a primitive polynomial invariant of \(G\) relative to \(L\). When possible, it is convenient to choose \(\Theta\) to be one of the polynomials in \((F)\). -- An algorithm is given that allows each polynomial \(F_i\) to be expressed as \(F_i({\mathbf X})= H_i(\Pi_1,\dots, \Pi_n,\Theta)\), an algebraic fraction in \(\Pi_1,\dots,\Pi_n\) and a polynomial in \(\Theta\). Now let \(L\) be the minimal polynomial of \(…
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 12Y05
Mathematics Subject Classification ID: 12D10
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 1025696
zbMATH Keywords: Cohen-Macaulay algebra
zbMATH Keywords: field theory
zbMATH Keywords: AXIOM
reviewed by: Francis Wright
describes a project that uses: AXIOM
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1025696