Puiseux parametrization


Theorem.

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

w=n=manzn/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