Is there a system $\{s_1, \cdots, s_m\}$ of symmetric polynomials of $z_1, \cdots, z_n \in \mathbb{C}$ such that
$s_1(z_1, \cdots, z_n) = c_1$ $s_2(z_1, \cdots, z_n) = c_2$ $\cdots$ $s_m(z_1, \cdots, z_n) = c_m$
has at most one solution $(z_1, \cdots, z_n)$ up to permutation, for all choices $c_1, \cdots, c_m \in \mathbb{C}$?
If so, what is the minimum value of $m$? And what are all such systems of polynomials $\{s_1, \cdots, s_m\}$ with this property?