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analytic algebraic function
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(Definition)
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Let  be a field, and let
 be the ring of convergent power series in  variables. An element in this ring can be thought of as a function defined in a neighbourhood of the origin in  to  . The most common cases for  are
 or
 , where the convergence is with respect to the standard euclidean metric. These definitions can also be generalized to other fields.
Definition 1 A function
 is said to be  -analytic algebraic if there exists a nontrivial polynomial
![$ p \in k[x_1,\ldots,x_n,y]$](http://images.planetmath.org:8080/cache/objects/7517/l2h/img11.png "$ p \in k[x_1,\ldots,x_n,y]$") such that
 for all  in a neighbourhood of the origin in  . If
 then  is said to be holomorphic algebraic and if
 then  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
 where  is a neighbourhood of the origin is said to be  -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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(view preamble)
| 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 4001 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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