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Branching theorem ：ウィキペディア英語版

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_ : 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.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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