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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.