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).
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$\begingroup$ Hm, you can solve the system using Gröbner bases. If it is symmetric, you might be able to do some tricks, to speed up the computation. If you can post a concrete example of $p$, I might be able to elaborate. I have looked a bit on common zeros of certain specializations of Schur polynomials (related: staff.math.su.se/shapiro/Articles/… ), and this type of things appear. $\endgroup$– Per AlexanderssonDec 17, 2015 at 1:28
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