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 $k\in \mathbb{\mathbb{N}}$ 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 .