inverse function of polynomials

Hi all,

I have a polynomial p(z), say of degree n; view it as an n-to-1 map of the complex Riemann sphere to itself. Can I always construct an inverse which is defined on large piece of the complex plane? of course I can define the inverse locally around each point where , but I want global inverses

example: z^2, its inverse is which we can define e.g. on

I hope to get that for any polynomial there exists a cut (or a few) of the complex plane such that the inverse is defined there globally

any ideas will be appreciated...