Puiseux parametrization


Suppose that VUC2 is an irreducible complex analyticPlanetmathPlanetmath subset of (complex) dimension 1 where U is a domain. Suppose that 0V. Then there exists an analytic (holomorphic) map f:DV, where D is the unit disc, such that f(0)=0 and f(D)=N where NV is a neighbourhood of 0 in V, f is one to one, and further f|D\{0} is a biholomorphism onto N\{0}. In fact there exist suitable local coordinates (z,w) in C2 such that f is then given by ξ(z,w) where z=ξk, w=n=manξn where m>k.

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.


  • 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