Computing critical points for invariant algebraic systems

Abstract

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