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 ${\displaystyle b:=f(a)\in Y}$. Then there exist $k\in \mathbb{N}$ and charts ${\displaystyle \psi _{1}:U_{1}\to V_{1}}$ on $X$ and ${\displaystyle \psi _{2}:U_{2}\to V_{2}}$ on $Y$ such that

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

This theorem gives rise to several definitions:

• We call $k$ the multiplicity of $f$ at $a$. Some authors denote this ${\displaystyle \nu (f,a)}$.
• 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.