In my book the following exercise is given: let $p(z)\in \mathbb{C}[z]$ be a complex polynomial with distinct roots and degree $n>1$. Determine the greatest neighborhood $V$ of 0 such that $p:p^{-1}(V)\to V$ is a covering map.
My attempt of solution: I've shown, using inverse function theorem, that, if $Y:=\{z\in \mathbb{C}: p'(z)=0\}$ and $U:=\mathbb{C}\setminus p^{-1}(p(Y))$, then $p:U \to p(U)$ is a covering map. Unfortunately there exist polynomials $p$ such that $p'(0)=0$. Any suggestions?
The problematic points for local inversion are the critical values of $p$, namely the points of the set $C = \{p(z) : p'(z) = 0\}$. Therefore, the largest open set covered by $p$ is $V = \mathbb C\setminus C$. We have $0\in V$ because if $0\in C$, that means $p(z)=0=p'(z)$ for some $z$, contradicting the assumption that the roots are distinct.