Analyticity of roots of a polynomial in terms of coefficients

Question

Suppose that $f(z,w)$ is a non-constant polynomial in $z,w$ with coefficients in $\mathbb{C}$.
Fix $z$, we define $p(w)=f(z,w)$. From Liouville's theorem, we know that $p(w)=0$ is solvable for $w$, and we choose a root $w=G(z)$.

It is known that $w=G(z)$ is a continuous function of $z$ because any root of a polynomial over $\mathbb{C}$ is continuous in terms of the coefficients.

I want to know: is it true that $w=G(z)$ is analytic in simply connected open regions excluding possible singularities and branch points ?

Answer 1

Yes, it is. Our numerical analysis teacher used to allude to it again and again. I think its name is after one the giant guy's name: Euler's theorem, Gauss theorem, ... Google it!