Solving a system of algebraic equations with symmetries
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 \(\Theta\) over \(k[{\mathbf X}]^L\); we have \(L({\mathbf X}, T)=\prod_{\Theta'\in L.\Theta}(T- \Theta')\in k[{\mathbf X}]^L[T]\) (where \(L\) is called a generic Lagrange resolvent). As \(k(\Pi_1,\dots, \Pi_n)= k({\mathbf X})^L\), we can write \(L({\mathbf X}, T)= H_0(\Pi_1,\dots,\Pi_n,T)\) where \(H_0\) is some rational function. The question \(H_0(\Pi_1,\dots,\Pi_n,\Theta)=0\) is always satisfied because \(\Theta\) is a root of \(L\). Then, we solve the system of \((p=1)\) algebraic equations \(H_i(\Pi_1,\dots, \Pi_n,\Theta)= 0\), \(0\leq i\leq p\) for \(\Pi_1,\dots,\Pi_n,\Theta\) as indeterminates. Theorem 1: Let \(D\in k[\Pi_1,\dots,\Pi_n]\) be the LCM of the denominators of all the fractions \(H_i\), \(0\leq i\leq p\) and let \(H_i'=DH_i\). For every solution \({\mathbf x}:= (x_1,\dots,x_n)\) of the system \((F): F_i({\mathbf X})= 0\), \(1\leq i\leq p\) there exists a solution \((\pi_1,\dots, \pi_n,\theta)\) of the system \((H'): H_i'(\Pi_1,\dots, \Pi_n,\Theta)= 0\), \(0\leq i\leq p\) such that \({\mathbf x}\) is a solution of the system \((P_\pi): \Pi_i({\mathbf X})= \pi_i\), \(1\leq i\leq n\) and of the equation \(\Theta({\mathbf X})=\theta\). Conversely, for any solution \((\pi_1,\dots, \pi_n,\theta)\) of the system \((H')\) such that \(D(\pi_1,\dots, \pi_n)\neq 0\), if \({\mathbf x}\) is a solution of the system \((P_\pi)\) relative to \((\pi_1,\dots, \pi_n)\), then there exists some \(A\in L\) such that \(\Theta(A.{\mathbf x})= \theta\), and then for all \(B\in G\), \(BA.{\mathbf x}\) is a soultion of the system \((F)\). A slighly more general version of this theorem is also given. The paper then presents an algorithm that applies the theory and has been implemented in AXIOM. It is followed by several examples.
scientific article; zbMATH DE number 1025696
scientific article