In mathematics, the branching theorem is a theorem about Riemann surfaces. Intuitively, it states that every non-constant holomorphic function is locally a polynomial.
Let X and Y be Riemann surfaces, and let f : X → Y be a non-constant holomorphic map. Fix a point a ∈ X and set b := f ( a ) ∈ Y . Then there exist k ∈ N and charts ψ 1 : U 1 → V 1 on X and ψ 2 : U 2 → V 2 on Y such that
ψ 1 ( a ) = ψ 2 ( b ) = 0 ; and ψ 2 ∘ f ∘ ψ 1 − 1 : V 1 → V 2 is z ↦ z k . This theorem gives rise to several definitions:
We call k the multiplicity of f at a . Some authors denote this ν ( 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.