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.

tatement of the theorem

Let $X$ and $Y$ be Riemann surfaces, and let $f\; :\; X\; o\; Y$ 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 $psi\_\{1\}\; :\; U\_\{1\}\; o\; V\_\{1\}$ on $X$ and $psi\_\{2\}\; :\; U\_\{2\}\; o\; V\_\{2\}$ on $Y$ such that
* $psi\_\{1\}\; (a)\; =\; psi\_\{2\}\; (b)\; =\; 0$; and
* $psi\_\{2\}\; circ\; f\; circ\; psi\_\{1\}^\{-1\}\; :\; V\_\{1\}\; o\; V\_\{2\}$ 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 $u\; (f,\; a)$.
* 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

*citation|first=Lars|last=Ahlfors|authorlink=Lars Ahlfors|title=Complex analysis|publisher=McGraw Hill|year=1953|publication-date=1979|edition=3rd|isbn=0-07-000657-1.