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



The holomorphic version of Galois theory
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.


