Puiseux parametrizationPuiseuxParametrization2005-06-16 19:22:402006-09-17 09:46:33Theorem% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{rmk}{Remark}\begin{thm}
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 $m > k$.
\end{thm}
This is sometimes written as
\begin{equation*}
w = \sum_{n=m}^\infty a_n z^{n/k}
\end{equation*}
and hence the name {\em 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 \cite{Chirka:CAS} page 98.
\begin{thebibliography}{9}
\bibitem{Chirka:CAS}
E.\@ M.\@ Chirka.
{\em \PMlinkescapetext{Complex Analytic Sets}}.
Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989.
\bibitem{Dimca:singu}
Alexandru Dimca.
{\em \PMlinkescapetext{Topics on Real and Complex Singularities}}.
Vieweg, Braunschweig, Germany, 1987.
\end{thebibliography}