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 and be Riemann surfaces, and let be a non-constant holomorphic map. Fix a point and set. Then there exist and charts on and on such that

; and
is

This theorem gives rise to several definitions:

We call the multiplicity

of at. Some authors denote this.

If, the point is called a branch point of.
If has no branch points, it is called unbranched. See also unramified morphism.