Slight generalization of the Local Normal Form

 

The Local Normal Form states that if F:X\to Y is a holomorphic map at p\in X, which is not constant, then there is a unique integer m\geq 1 which satisfies the following property: for every chart \phi_2: U_2\to V_2 on Y centered at F(p), there exists a chart \phi_1: U_i\to V_1 on X centered at p such that \phi_2(F(\phi_1^{-1}(z)))=z^m.

The way I understand the proof, I think we can extend it to say that for any chart U_1\to V_1 on X centered at p, then there exists a chart \phi_2: U_2\to V_2 on Y centered at F(p) such that \phi_2(F(\phi_1^{-1}(z)))=z^m. The only added condition is that \phi_2: U_2\to V_2 is non-zero except at F(p).

Proving this would be quite easy, and is left as an exercise.

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