The thing you are asking was much studied in connection with Hilbert Problem 13.
The roots of a polynomial of degree exactly $d$ form an unordered $d$-tuple. The set of
unordered $d$-tuples is called the configuration space. It is the factor of $C^d$
 over
the action of permutation group. It is equivalent to the space of
polynomials of degree exactly $d$ modulo multiplication by a non-zero constant.
One recent reference is http://arxiv.org/pdf/math/0403120v3, and it contains many other references. The version of Hilbert problem 13 asks whether this function, mapping a polynomial to its roots, can be represented as a composition of functions of fewer variables.