analytic algebraic function

Let k be a field, and let k{x1,,xn} 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 kn to k. The most common cases for k are or , where the convergence is with respect to the standard euclidean metricMathworldPlanetmath. These definitions can also be generalized to other fields.


A function fk{x1,,xn} is said to be k-analytic algebraic if there exists a nontrivial polynomial pk[x1,,xn,y] such that p(x,f(x))0 for all x in a neighbourhood of the origin in kn. If k= then f is said to be holomorphic algebraic and if k= 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.


A mapping f:Uknkm 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. , 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