Computing critical points for invariant algebraic systems

Let K be a field and (f1,…,fs,ϕ) multivariate polynomials in K[x1,…,xn] (with s<n) each invariant under the action of Sn, group permutations {1,…,n}. We consider problem computing critical points ϕ restricted to algebraic set V(f), where f=(f1,…,fs). This is same as at which f vanishes Jacobian matrix associated rank deficient, provided that this finite. exploit invariance properties input split solution space according orbits Sn. allows us design an algorithm gives triangular description runs time polynomial ds, (n+dd) (ns+1) d maximum degree polynomials. When d,s are fixed, n while when s fixed d≃n yields exponential speed-up with respect usual system solving algorithms.