Suppose $F(z,w)=0$, where $F$ is a polynomial in both variables, further suppose that we have a taylor series for $w(z)$ in a neighborhood of a point $z_0$ : $$ w(z)=w_0 + \sum_{k=1}^{\infty} a_k (z-z_0)^k $$
also suppose that $\displaystyle \frac{\partial F}{\partial z} \Big|_{(z_0,w_0)} \neq 0 $; the implicit function theorem tells us that we can solve for $z$ in a sufficiently small neighborhood of $z_0$; but do we necessarily know that w(z) would be a biholomorphic map ? i.e. is it correct that $$z-z_0 = \sum^{\infty}_{k=1} b_k(w-w_0)^k$$ with $\displaystyle b_1=\frac{1}{a_1}$.
Any help is appreciated, thanks
Yes, there is an analytic implicit function theorem. And if $w(z)$ is analytic near $z_0$ with $w'(z_0) \ne 0$, then $w$ is one-to-one in a neighbourhood of $z_0$, its inverse on the image of that neighbourhood being analytic.