Puiseux parametrization
Theorem.
Suppose that is an irreducible complex analytic subset of (complex) dimension 1 where is a domain. Suppose that . Then there exists an analytic (holomorphic) map , where is the unit disc, such that and where is a neighbourhood of in , is one to one, and further is a biholomorphism onto . In fact there exist suitable local coordinates in such that is then given by where , where .
This is sometimes written as
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.
References
- 1 E. M. Chirka. . Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989.
- 2 Alexandru Dimca. . Vieweg, Braunschweig, Germany, 1987.
Title | Puiseux parametrization |
Canonical name | PuiseuxParametrization |
Date of creation | 2013-03-22 15:20:32 |
Last modified on | 2013-03-22 15:20:32 |
Owner | jirka (4157) |
Last modified by | jirka (4157) |
Numerical id | 6 |
Author | jirka (4157) |
Entry type | Theorem |
Classification | msc 32B10 |
Synonym | Puiseux series parametrization |
Synonym | Puiseux normalization |
Synonym | Puiseux series normalization |
Synonym | Puiseux parameterization |
Synonym | Puiseux series parameterization |
Related topic | PuiseuxSeries |