mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the branching theorem is a theorem about Riemann surfaces. Intuitively, it states that every non-constant

holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...

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

chart A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent tabu ...

\psi_ : U_ \to V_

X

and

\psi_ : U_ \to V_

Y

such that *

\psi_ (a) = \psi_ (b) = 0

; and *

\psi_ \circ f \circ \psi_^ : V_ \to V_

z \mapsto z^.

This theorem gives rise to several definitions: * We call

k

the '' multiplicity '' of

f

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

*. Theorems in complex analysis Riemann surfaces {{Riemannian-geometry-stub