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 $$k \in \N$$ 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: ]]'' of $$f$$ at $$a$$. Some authors denote this $$\nu (f, a)$$.
 * We call $$k$$ the ''[[Multiplicity (mathematics)|multiplicity
 * 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.