Puiseux parametrization

Theorem.

Suppose that $V\mathrm{\subset }U\mathrm{\subset }{\mathrm{C}}^{\mathrm{2}}$ is an irreducible complex analytic subset of (complex) dimension 1 where $U$ is a domain. Suppose that $\mathrm{0}\mathrm{\in }V$. Then there exists an analytic (holomorphic) map $f\mathrm{:}\mathrm{D}\mathrm{\to }V$, where $\mathrm{D}$ is the unit disc,such that $f\mathit{}\mathrm{(}\mathrm{0}\mathrm{)}\mathrm{=}\mathrm{0}$ and $f\mathit{}\mathrm{(}\mathrm{D}\mathrm{)}\mathrm{=}N$ where $N\mathrm{\subset }V$ is a neighbourhood of $\mathrm{0}$ in $V$, $f$ is one to one, and further ${f\mathrm{|}}_{\mathrm{D}\mathrm{\backslash \backslash }\mathrm{\{}\mathrm{0}\mathrm{\}}}$ is a biholomorphism onto $N\mathrm{\backslash \backslash }\mathrm{\{}\mathrm{0}\mathrm{\}}$.In fact there exist suitable local coordinates $\mathrm{(}z\mathrm{,}w\mathrm{)}$ in ${\mathrm{C}}^{\mathrm{2}}$ such that $f$ is then given by $\xi \mathrm{\mapsto }\mathrm{(}z\mathrm{,}w\mathrm{)}$ where $z\mathrm{=}{\xi }^k$,$w\mathrm{=}{\mathrm{\sum }}_{n\mathrm{=}m}^{\mathrm{\infty }}{a}_n\mathit{}{\xi }^n$ where $m\mathrm{>}k$.

This is sometimes written as

$$w=\sum _{n=m}^{\mathrm{\infty }}{a}_n{z}^{n/k}$$

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.