Computing critical points for invariant algebraic systems

Let \(\mathbf{K}\) be a field and \(\phi\), \(\mathbf{f} = (f_1, \ldots, f_s)\) in \(\mathbf{K}[x_1, \dots, x_n]\) be multivariate polynomials (with \(s < n\)) invariant under the action of \(\mathcal{S}_n\), the group of permutations of \(\{1, \dots, n\}\). We consider the problem of computing the points at which \(\mathbf{f}\) vanish and the Jacobian matrix associated to \(\mathbf{f}, \phi\) is rank deficient provided that this set is finite. We exploit the invariance properties of the input to split the solution space according to the orbits of \(\mathcal{S}_n\). This allows us to design an algorithm which gives a triangular description of the solution space and which runs in time polynomial in \(d^s\), \({{n+d}\choose{d}}\) and \(\binom{n}{s+1}\) where \(d\) is the maximum degree of the input polynomials. When \(d,s\) are fixed, this is polynomial in \(n\) while when \(s\) is fixed and \(d \simeq n\) this yields an exponential speed-up with respect to the usual polynomial system solving algorithms.