Ramification index, ramification order and branch value

Let $f:X\to Y$ be a nonconstant holomorphic map between Riemann surfaces and let $x\in X$ (Holomorphic maps and meromorphic functions on Riemann surfaces). By Local power-map normal form on Riemann surfaces there are holomorphic coordinate charts $\phi$ centred at $x$ and $\psi$ centred at $f(x)$ such that the chart expression is the power map $\psi \circ f\circ {\phi }^{-1}(z)=z^e\text{for }z\text{ near }0,$ with a unique positive integer $e$. Define:

1. the ramification index of $f$ at $x$ to be this exponent, $e_x(f):=e\ge 1;$
2. the ramification order of $f$ at $x$ to be $e_x(f)-1\ge 0$;
3. $x$ to be a critical point (or ramification point) of $f$ when $e_x(f)>1$, and $f$ to be unramified at $x$ when $e_x(f)=1$;
4. a branch value of $f$ to be a point $y\in Y$ for which there is a critical point $x\in X$ with $f(x)=y$; the set of branch values is the branch locus of $f$, and the set of critical points is the critical locus.

Conventions. The index is well defined because the exponent of the normal form is unique; the proof below records this together with the equivalent descriptions $e_x(f)={\mathrm{deg}}_x(\psi \circ f\circ {\phi }^{-1})={\mathrm{ord}}_x(f-f(x))$ in any centred charts (Local degree of a nonconstant holomorphic map), and $e_x(f)=1$ exactly when $f$ is a local biholomorphism at $x$ (Biholomorphic maps between complex domains). In particular a critical point is a point where $f$ is not locally injective, and the set of critical points is discrete in $X$. A meromorphic function on $X$ is a holomorphic map to the Riemann sphere (Holomorphic maps and meromorphic functions on Riemann surfaces), so the same index, order and branch language applies to it; a pole of a meromorphic function is a point where its value is the point at infinity, and says nothing by itself about ramification. No choice principle is used anywhere in this definition: an index is a single positive integer determined by local data.

The index $e_x(f)$ is the multiplicity used in Degree of a proper holomorphic map of Riemann surfaces, where a weighted fibre count ${\sum }_{x\in f^{-1}(y)}{e}_x(f)$ is shown to be independent of $y$, and in Riemann–Hurwitz formula for compact Riemann surfaces, where the numbers $e_x(f)-1$ are summed over the critical locus. Both uses require the finiteness of the critical locus on a compact surface and not merely its discreteness; that finiteness is a consequence of compactness and is stated and used where it is needed. The local normal form theorem is the only place where the exponent is manufactured, and it is applied to a nonconstant map throughout; a constant map has no honest local power form and is excluded by hypothesis.