Analyticity of roots of a polynomial in terms of coefficients 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 ?
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!
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!