Branching theorem explained

Branching theorem should not 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 \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

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