Solving polynomial systems globally invariant under an action of the symmetric group and application to the equilibria of N vortices in the plane
Jean‐Charles Faugère, Jules Svartz · 2012
We propose an efficient algorithm to solve polynomial systems of which equations are globally invariant under an action of the symmetric group GN acting on the variable xi with σ(xi) = xσ(i) and the number of variables is a multiple of N. For instance, we can assume that swapping two variables (or two pairs of variables) in one equation gives rise to another equation of the system (perhaps changing the sign). The idea is to apply many times divided difference operators to the original system in order to obtain a new system of equations involving only the symmetric functions of a subset of the variables.