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 $k\in \mathbb{N}$ and charts $\psi _{1}:U_{1}\to V_{1}$ on $X$ and $\psi _{2}:U_{2}\to V_{2}$ on $Y$ such that

$\psi _{1}(a)=\psi _{2}(b)=0$ ; and
$\psi _{2}\circ f\circ \psi _{1}^{-1}:V_{1}\to V_{2}$ 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 $\nu (f,a)$ .
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.

References

Ahlfors, Lars (1953), Complex analysis (3rd ed.), McGraw Hill (published 1979), ISBN 0-07-000657-1 .

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