Let $ P(z) $ be a $\textit{formal}$ power series in $z$ that a priori may not have a non zero radius of convergence. Assume that $P(0) =0$. 


Let $\Phi(w,z)$ be a polynomial in two variables, that is not identically zero. Assume that 
$\Phi(0,0) =0$. Suppose $\textbf{formally}$ we have the identity 


$$ \Phi(P(z),z) =0 $$ 


Can we conclude that $P(z)$ has a non zero radius of convergence? 


Everything is over the complex numbers $\mathbb{C}$. 



If  a formal power series over the complex numbers satisfies a polynomial identity, does it imply that the power series has a radius of convergence?
The equation $\Phi(w,z)=0$ can be solved using Puiseux series. If $\frac{\partial{\Phi}}{\partial{w}}\not\equiv 0$ then there exist finitely many formal series $f(z)=\sum\_{n\geq0}a\_nz^{n/p}$ such that formally $\Phi(w,z)=0$. All these series
are convergent. So the answer to your question is positive.


For the proof see any book titled "Algebraic functions".


