Local Normal Form of holomorphic map on Riemann surface

Question

Let $F:X \rightarrow Y $ be a holomorphic map defined at $p\in X$ , which is not constant. Then there is a unique integer $m \geq1$ which satisfies the following property: for every chart $\Phi_2:U_2 \rightarrow V_2$ on $Y$ centered at $F(p)$ , there there exists a chart $\Phi_1: U_1 \rightarrow V_1$ centered at $p$ such that $\Phi_2(F(\Phi_1^{-1}(z)))=z^m $. The existence part of the question is fine . How to prove uniqueness of $m\geq1$? Reference ( Algebraic Curves and Riemann surfaces by Rick Miranda).

Answer 1

I am trying to give a proof of uniqueness of the integer $m\geq1$ . Verify if I am wrong. First observe that if there are local coordinate charts near p and F(p) such that F has the form $z\in z^m$ then there are exactly m preimages of points near $F(p)$ in a neighborhood of $p$ . As $F$ is an open mapping therefore $F(U_1) \cap U_2$ is open in Y , $ g =\Phi_2(F(\Phi_1^{-1}(z)))$ is holomorphic map which is again open map. Therefore we can get a small enough neighborhood of $F(p)$ which has exactly m inverse images ,can be easily verified via commutative diagram. // Therefore if we get two pairs of chart around $p$ and $F(p)$ for which $F$ locally looks like $z\rightarrow z^m $ and $z\rightarrow z^n $ respectively , $m \neq n$ then we will get a neighborhood around $p$ and $F(p)$ where some point has once $m$ preimages and $n $ preimages simultaneously. Which is absurd . therefore $m=n$.