TITLE: The holomorphic version of Galois theory
QUESTION [9 upvotes]: We identify the space of polynomials of degree n with $\mathbb{C}^{n+1}-\mathbb{C}^{n}$, that is an $n+1$ tuple $(a_{n},a_{n-1},\ldots,a_{0})$  with $a_{n} \neq 0$ is identified with $p(z)=a_{n}z^{n} +\ldots a_{1}z+a_{0}$. In this question we search for a holomorphic representation for the roots of P. That is, we search for holomorphic maps which send each polynomial $P$ to one of its roots. Motivating by the special case $n=2$ and the "Radical formula" for the roots,  it is natural to search for  an appropriate version of "Riemann surface of radical-type functions". So here is our question, precisely:
Question:

Does there exist an $n+1$ dimensional  complex manifold M  with  a covering space structure $\pi \colon M \rightarrow  \mathbb{C}^{n+1}-\mathbb{C}^{n}$ and  a holomorphic  function $f \colon M \rightarrow \mathbb{C}^{n}$, such that for every $\tilde{P} \in M$ with $\pi(\tilde{P})=P$, all $n$ roots of $P$ are arranged in the $n$-tuple $f(\tilde{P})$?

Remark:
Since "Galois Theory " is  an obstruction for existence  of a radical (algebraic) formulation for the roots, it was natural that we search for  a holomorphic analogy

REPLY [11 votes]: 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.