Let $f(z)=\sum_{n\ge0}a_nz^n$ be a power series defined on a open subset $\mathcal U$ of $\mathbb C$. I want to prove that if $f$ is algebraic over $\mathbb C(z)$, that is there exists non all zero polynomials $P_0,\cdots,P_s$ of $\mathbb C[z]$ such that for every $z\in\mathcal U$, $\sum_{i=0}^sP_i(z)f^i(z)=0$, then $f$ admits only finitely many poles and zeros.
But I did not manage to prove that. Any answer will be welcome.
Thanks in advance