analytic algebraic function
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.
A function is said to be -analytic algebraic if there exists a nontrivial polynomial 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.
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. , Princeton University Press, Princeton, New Jersey, 1999.
|Title||analytic algebraic function|
|Date of creation||2013-03-22 15:36:05|
|Last modified on||2013-03-22 15:36:05|
|Last modified by||jirka (4157)|
|Synonym||-analytic algebraic function|
|Defines||holomorphic algebraic function|
|Defines||real-analytic algebraic function|
|Defines||analytic algebraic mapping|