symmetric systems of polynomial equations

Suppose I have a polynomial p(x1,...,xN) in N complex variables, and I wish to solve p(xπ(1),...,xπ(N))=0 for all permutations πSN. 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).

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