Branching theorem

Not to be confused with Branch theory.

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 ${\displaystyle X}$ and ${\displaystyle Y}$ be Riemann surfaces, and let ${\displaystyle f:X\to Y}$ be a non-constant holomorphic map. Fix a point ${\displaystyle a\in X}$ and set ${\displaystyle b:=f(a)\in Y}$. Then there exist ${\displaystyle k\in \mathbb {N} }$ and charts ${\displaystyle \psi _{1}:U_{1}\to V_{1}}$ on ${\displaystyle X}$ and ${\displaystyle \psi _{2}:U_{2}\to V_{2}}$ on ${\displaystyle Y}$ 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 ${\displaystyle k}$ the multiplicity of ${\displaystyle f}$ at ${\displaystyle a}$. Some authors denote this ${\displaystyle \nu (f,a)}$.
• If ${\displaystyle k>1}$, the point ${\displaystyle a}$ is called a branch point of ${\displaystyle f}$.
• If ${\displaystyle f}$ has no branch points, it is called unbranched. See also unramified morphism.

References

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