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.
Categories: cs.SC