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
