PlanetMath: Puiseux parametrization

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (22)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

Puiseux parametrization

(Theorem)

Theorem 1 Suppose that $V \subset U \subset {\mathbb{C}}^2$ is an irreducible complex analytic subset of (complex) dimension 1 where is a domain. Suppose that $0 \in V$ . Then there exists an analytic (holomorphic) map $f \colon {\mathbb{D}} \to V$ , where ${\mathbb{D}}$ is the unit disc, such that and $f({\mathbb{D}}) = N$ where $N \subset V$ is a neighbourhood of 0 in , is one to one, and further $f \vert _{{\mathbb{D}}\backslash \{0\}}$ is a biholomorphism onto $N \backslash \{0\}$ . In fact there exist suitable local coordinates in ${\mathbb{C}}^2$ such that is then given by $\xi \mapsto (z,w)$ where $z = \xi^k$ , $w = \sum_{n=m}^\infty a_n \xi^n$ where .

This is sometimes written as

$\displaystyle w = \sum_{n=m}^\infty a_n z^{n/k}$

and hence the name Puiseux series parametrization. If you do however write it like this, it must be properly interpreted, as the Puiseux series is in general not single valued.

A similar result for arbitrary complex analytic sets with singularities of codimension 1 in higher dimensional spaces under further conditions on the singular set was obtained by Stutz, see Chirka [1] page 98.

Bibliography

1: E. M. Chirka. Complex Analytic Sets. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989.
2: Alexandru Dimca. Topics on Real and Complex Singularities. Vieweg, Braunschweig, Germany, 1987.

"Puiseux parametrization" is owned by jirka.

(view preamble | get metadata)