Branching theorem

In mathematics, the branching theorem is a theorem about Riemann surfaces. Intuitively, it states that every non-constant holomorphic function is locally a polynomial.

Statement of the theorem

Let $X$ and $Y$ be Riemann surfaces, and let $f : X \to Y$ be a non-constant holomorphic map. Fix a point $a \in X$ and set $b := f (a) \in Y$ . Then there exist $k \in ℕ$ and charts $ψ_{1} : U_{1} \to V_{1}$ on $X$ and $ψ_{2} : U_{2} \to V_{2}$ on $Y$ such that

$ψ_{1} (a) = ψ_{2} (b) = 0$ ; and
$ψ_{2} \circ f \circ ψ_{1}^{- 1} : V_{1} \to V_{2}$ is $z \mapsto z^{k} .$

This theorem gives rise to several definitions:

We call $k$ the multiplicity of $f$ at $a$ . Some authors denote this $ν (f, a)$ .
If $k > 1$ , the point $a$ is called a branch point of $f$ .
If $f$ has no branch points, it is called unbranched. See also unramified morphism.