real analytic subvariety

Let UN be an open set.


A closed set XU is called a real analytic subvariety of U such that for each point pX, there exists a neigbourhood V and a set of real analytic functions defined in V, such that

XV={pVf(p)=0 for all f}.

If U=N and all the f are real polynomials, then X is said to be a real algebraic subvariety.

If X is not required to be closed, then it is said to be a local real analytic subvariety. Sometimes X is called a real analytic set or real analytic variety. Similarly as for complex analytic sets we can also define the regular and singular pointsMathworldPlanetmathPlanetmath.


A point pX is called a regular point if there is a neighbourhood V of p such that XV is a submanifoldMathworldPlanetmath. Any other point is called a singular point.

The set of regular points of X is denoted by X- or sometimes X*. The set of singular points is no longer a subvariety as in the complex case, though it can be sown to be semianalytic. In general, real subvarieties is far worse behaved than their complex counterparts.


  • 1 Jacek Bochnak, Michel Coste, Marie-Francoise Roy. . Springer, 1998.
Title real analytic subvariety
Canonical name RealAnalyticSubvariety
Date of creation 2013-03-22 17:41:07
Last modified on 2013-03-22 17:41:07
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 4
Author jirka (4157)
Entry type Definition
Classification msc 14P05
Classification msc 14P15
Synonym real analytic variety
Synonym real analytic set
Related topic SmoothSubmanifoldContainedInASubvarietyOfSameDimensionIsRealAnalytic
Defines real algebraic variety
Defines real algebraic subvariety
Defines local real analytic subvariety
Defines regular point
Defines singular point