analytic algebraic function

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.

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.

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.

References

1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.

Title	analytic algebraic function
Canonical name	AnalyticAlgebraicFunction
Date of creation	2013-03-22 15:36:05
Last modified on	2013-03-22 15:36:05
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	7
Author	jirka (4157)
Entry type	Definition
Classification	msc 14-00
Classification	msc 14P20
Synonym	$k$ -analytic algebraic function
Synonym	analytic algebraic
Defines	holomorphic algebraic function
Defines	real-analytic algebraic function
Defines	Nash function
Defines	analytic algebraic mapping