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analytic algebraic function
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(Definition)
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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 $k$ . The most common cases for $k$ are $\mathbb{C}$ or $\mathbb{R}$ , where the convergence is with respect to the standard euclidean metric. These definitions can also be generalized to other fields.
Definition 1 A function $f \in k\{x_1,\ldots,x_n\}$ is said to be $k$ -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 holomorphic algebraic and if $k=\mathbb{R}$ then $f$
is said to be real-analytic algebraic or a Nash function.
The same definition applies near any other point other then the origin by just translation.
Definition 2 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.
- 1
- M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. Real Submanifolds in Complex Space and Their Mappings, Princeton University Press, Princeton, New Jersey, 1999.
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"analytic algebraic function" is owned by jirka.
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| Other names: |
 -analytic algebraic function, analytic algebraic |
| Also defines: |
holomorphic algebraic function, real-analytic algebraic function, Nash function, analytic algebraic mapping |
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Cross-references: component, mapping, translation, point, near, holomorphic, polynomial, algebraic, definitions, Euclidean metric, origin, neighbourhood, function, variables, power series, convergent, ring, field
There is 1 reference to this entry.
This is version 4 of analytic algebraic function, born on 2005-12-05, modified 2005-12-09.
Object id is 7517, canonical name is AnalyticAlgebraicFunction.
Accessed 5573 times total.
Classification:
| AMS MSC: | 14-00 (Algebraic geometry :: General reference works ) | | | 14P20 (Algebraic geometry :: Real algebraic and real analytic geometry :: Nash functions and manifolds) |
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Pending Errata and Addenda
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