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'analytic algebraic function'
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| Title of object: |
analytic algebraic function |
| Canonical Name: |
AnalyticAlgebraicFunction |
| Type: |
Definition |
| Created on: |
2005-12-05 20:51:43 |
| Modified on: |
2005-12-07 10:39:14 |
| Classification: |
msc:14-00, msc:14P20 |
| Defines: |
holomorphic algebraic function, real-analytic algebraic function, Nash function, analytic algebraic mapping |
| Synonyms: |
analytic algebraic function=$k$-analytic algebraic function
analytic algebraic function=analytic algebraic |
Revision comment (for changes between this and next version):
| Changes for correction #7308 ('Metric on k?'). |
Preamble:
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Content:
Let $k$ be a field, and let $k\{x_1,\ldots,x_n\}$ be the ring of convergent
power series in $n$ variables. An element in this ring can be thought of as
a function defined in a neighbourhood of the origin in $k^n$ to $k$.
\begin{defn}
A function $f \in k\{x_1,\ldots,x_n\}$ is said to be \emph{$k$-analytic
algebraic} if there exists a nontrivial polynomial $p \in
k[x_1,\ldots,x_n,y]$ such that $p(x,f(x)) \equiv 0$ for all $x$ in a
neighbourhood of the origin in $k^n$.
If $k=\mathbb{C}$ then $f$ is said to be \emph{holomorphic algebraic} and if
$k=\mathbb{R}$ then $f$ is said to be \emph{real-analytic algebraic} or a
\emph{Nash function}.
\end{defn}
The same definition applies near any other point other then the origin by just translation. The field need not be $\mathbb{C}$ or $\mathbb{R}$ but those are the most common cases.
\begin{defn}
A mapping $f \colon U \subset k^n \to k^m$ where $U$ is a neighbourhood of the origin is said to be $k$-analytic algebraic if each component function is analytic algebraic.
\end{defn}
\begin{thebibliography}{9}
\bibitem{ber:submanifold}
M.\@ Salah Baouendi,
Peter Ebenfelt,
Linda Preiss Rothschild.
{\em \PMlinkescapetext{Real Submanifolds in Complex Space and Their Mappings}},
Princeton University Press,
Princeton, New Jersey, 1999.
\end{thebibliography} |
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