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Puiseux parametrization
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(Theorem)
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Theorem 1 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 \begin{equation*} w = \sum_{n=m}^\infty a_n z^{n/k} \end{equation*}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. Complex Analytic Sets. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989.
- 2
- Alexandru Dimca. Topics on Real and Complex Singularities. Vieweg, Braunschweig, Germany, 1987.
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"Puiseux parametrization" is owned by jirka.
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See Also: Puiseux series
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Puiseux series parametrization, Puiseux normalization, Puiseux series normalization, Puiseux parameterization, Puiseux series parameterization |
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Cross-references: singular set, codimension, similar, single valued, Puiseux series, local coordinates, onto, neighbourhood, unit disc, map, holomorphic, domain, dimension, complex, subset, complex analytic, irreducible
There are 2 references to this entry.
This is version 3 of Puiseux parametrization, born on 2005-06-16, modified 2006-09-17.
Object id is 7162, canonical name is PuiseuxParametrization.
Accessed 4435 times total.
Classification:
| AMS MSC: | 32B10 (Several complex variables and analytic spaces :: Local analytic geometry :: Germs of analytic sets, local parametrization) |
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Pending Errata and Addenda
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