## 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 π∈SN. 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. … Read more