Solving a System of Algebraic Equations with Symmetries
Résumé
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 invariants $\Pi_1,\cdots,\Pi_n$ (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 $k(\Pi_1,\cdots,\Pi_n)$. 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 $(x_1,\cdots,x_n)$ breaks the field extension $k(x_1,\cdots,x_n)$: $k$ in two parts (or more).