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 {\displaystyle X} and Y {\displaystyle Y} be Riemann surfaces, and let f : X → Y {\displaystyle f:X\to Y} be a non-constant holomorphic map. Fix a point a ∈ X {\displaystyle a\in X} and set b := f ( a ) ∈ Y {\displaystyle b:=f(a)\in Y} . Then there exist k ∈ N {\displaystyle k\in \mathbb {N} } and charts ψ 1 : U 1 → V 1 {\displaystyle \psi _{1}:U_{1}\to V_{1}} on X {\displaystyle X} and ψ 2 : U 2 → V 2 {\displaystyle \psi _{2}:U_{2}\to V_{2}} on Y {\displaystyle Y} such that
- ψ 1 ( a ) = ψ 2 ( b ) = 0 {\displaystyle \psi _{1}(a)=\psi _{2}(b)=0} ; and
- ψ 2 ∘ f ∘ ψ 1 − 1 : V 1 → V 2 {\displaystyle \psi _{2}\circ f\circ \psi _{1}^{-1}:V_{1}\to V_{2}} is z ↦ z k . {\displaystyle z\mapsto z^{k}.}
This theorem gives rise to several definitions:
- We call k {\displaystyle k} the multiplicity of f {\displaystyle f} at a {\displaystyle a} . Some authors denote this ν ( f , a ) {\displaystyle \nu (f,a)} .
- If k > 1 {\displaystyle k>1} , the point a {\displaystyle a} is called a branch point of f {\displaystyle f} .
- If f {\displaystyle f} has no branch points, it is called unbranched. See also unramified morphism.
- Ahlfors, Lars (1953), Complex analysis (3rd ed.), McGraw Hill (published 1979), ISBN 0-07-000657-1.