Solving a system of algebraic equations with symmetries Original Research Article

We propose a method to solve some polynomial systems whose equations are invariant by the action of a finite matrix multiplicative group G. It consists of expressing the polynomial equations in terms of some primary invariantsΠ1,…,Πn (e.g., the elementary symmetric polynomials), and one single “primitive” secondary invariant. The primary invariants are a transcendence basis of the algebra of invariants of 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 vector space over k(Π1,h.,Πn). The solutions of the system are given as roots of polynomials whose coefficients themselves are given as roots of some other polynomials: the representation of the solutions (x1,…,xn) breaks the field extension k(x1,…,xn): k in two parts (or more).