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 $X$ 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 (1979), Complex analysis (3rd ed.), McGraw Hill, ISBN 0-07-000657-1.