I'll call a polynomial in $z_1,..,z_N$ cyclic if it is invariant under cyclic permutation of the indices. I hope that's standard terminology.

I have N complex numbers $(z_1,...,z_N)$. I want to be able to compute what they are up to cyclic permutation, given the value of some set of cyclic polynomials. For example, if $N=2$ and I know the values of $z_1+z_2$ and of $z_1z_2$, then I could compute $(z_1,z_2)$ up to cyclic permutation. I imagine that there is a solution for general $N$. I also imagine that there is an algorithm that finds the solution from the given finite set of values. Floating point accuracy is good enough for me.

By using the symmetric polynomials, I should be able to find the $z_i$ except for an arbitrary permutation of the indices, but that's not good enough. (I use the word "should" because I don't know how one makes sure that some algorithm like Newton's method actually converges---one needs a good initial point.)

It would be nice to be able to do this with exactly N cyclic polynomials, as in the case N=2. Is the problem any easier if you are allowed to use polynomials in both $z_i$ and its conjugate $\overline{z_i}$: they still have to be invariant under cyclic permutation of the indices?