Computing Critical Points for Algebraic Systems Defined by Hyperoctahedral Invariant Polynomials

Let K be a field of characteristic zero and K[x1,...,xn] the corresponding multivariate polynomial ring. Given a sequence of s polynomials f = (f_1,...,f_s) and a polynomial φ, all in K[x1,...,xn] with s>n, we consider the problem of computing the set W(φ,f ) of points at which f vanishes and the Jacobian matrix of f, φ with respect to x1,...,xn does not have full rank. This problem plays an essential role in many application areas.