SOLVING A SYSTEM OF ALGEBRAIC EQUATIONS WITH SYMMETRIES

We propose a method to solve some polynomial systems whose equations a re invariant by the action of a finite matrix multiplicative group G. It consists of expressing the polynomial equations in terms of some pr imary invariants Pi(1), ..., Pi(n) (e.g., the elementary symmetric pol ynomials), and one single ''primitive'' secondary invariant. The prima ry invariants are a transcendence basis of the algebra of invariants o f the group G over the ground field k, and the powers of the primitive invariant give a basis of the field of invariants considered as a vec tor space over k(Pi(1), ..., Pi(n)). The solutions of the system are g iven as roots of polynomials whose coefficients themselves are given a s roots of some other polynomials: the representation of the solutions (x(1), ..., x(n)) breaks the field extension k(x(1), ..., x(n)) : k i n two parts (or more). (C) 1997 Published by Elsevier Science B.V.