Abstract: Let f(x) = \sum a_i x^i be a monic polynomial of degree n whose coefficients are algebraically independent variables over a base field k of characteristic 0. We say that a polynomial g(x) is generating (for the symmetric group) if it can be obtained from f(x) by a non-degenerate Tschirnhaus transformation. We show that the minimal number d_k(n) of algebraically independent coefficients of such a polynomial is at least [n/2]. This generalizes a classical theorem of Felix Klein on quintic polynomials and is related to an algebraic form of Hilbert's 13-th problem.
Our approach to this question (and generalizations) is based on the idea of the ``essential dimension'' of a finite group G: the smallest possible dimension of an algebraic G-variety over k to which one can ``compress'' a faithful linear representation of G. We show that d_k(n) is just the essential dimension of the symmetric group S_n. We give results on the essential dimension of other groups. In the last section we relate the notion of essential dimension to versal polynomials and discuss their relationship to the generic polynomials of Kuyk, Saltman and DeMeyer.