# symmetric systems of polynomial equations

Suppose I have a polynomial $p(x_1,...,x_N)$ in $N$ complex variables, and I wish to solve $p(x_{\pi(1)},...,x_{\pi(N)})=0$ for all permutations $\pi \in S_N$. Clearly this is overdetermined for generic $p$, but suppose $p$ is symmetric under exchange of all but one variable. Then this gives $N$ distinct equations, and so generically one expects a discrete set of solutions. Are there any general techniques for solving such a system of equations? I’m most interested in simply counting the number of solutions (up to permutations).