# Puiseux parametrization

###### Theorem.

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

This is sometimes written as

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

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.

## 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