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 try to give a holomorphic representation for the roots of P. That is holomorphic maps which send  each P to its roots. Motivating by the special case n=2 and the "Radical formula" it is natural to search for  an appropriate Riemann surface. So our explicit question is the following :

 Question:

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