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 [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] be Riemann surfaces, and let [math]\displaystyle{ f : X \to Y }[/math] be a non-constant holomorphic map. Fix a point [math]\displaystyle{ a \in X }[/math] and set [math]\displaystyle{ b := f(a) \in Y }[/math]. Then there exist [math]\displaystyle{ k \in \N }[/math] and charts [math]\displaystyle{ \psi_{1} : U_{1} \to V_{1} }[/math] on [math]\displaystyle{ X }[/math] and [math]\displaystyle{ \psi_{2} : U_{2} \to V_{2} }[/math] on [math]\displaystyle{ Y }[/math] such that

• [math]\displaystyle{ \psi_{1} (a) = \psi_{2} (b) = 0 }[/math]; and
• [math]\displaystyle{ \psi_{2} \circ f \circ \psi_{1}^{-1} : V_{1} \to V_{2} }[/math] is [math]\displaystyle{ z \mapsto z^{k}. }[/math]

This theorem gives rise to several definitions:

• We call [math]\displaystyle{ k }[/math] the multiplicity of [math]\displaystyle{ f }[/math] at [math]\displaystyle{ a }[/math]. Some authors denote this [math]\displaystyle{ \nu (f, a) }[/math].
• If [math]\displaystyle{ k \gt 1 }[/math], the point [math]\displaystyle{ a }[/math] is called a branch point of [math]\displaystyle{ f }[/math].
• If [math]\displaystyle{ f }[/math] 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 .