real analytic subvariety

Let $U\subset {ℝ}^N$ be an open set.

Definition.

A closed set $X\subset U$ is called a real analytic subvariety of $U$ such that for each point $p\in X$, there exists a neigbourhood $V$ and a set $ℱ$ of real analytic functions defined in $V$, such that

$$X\cap V=\{p\in V\mid f(p)=0\text{ for all }f\in ℱ\}.$$

If $U={ℝ}^N$ and all the $f\in ℱ$ 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 points.

Definition.

A point $p\in X$ is called a regular point if there is a neighbourhood $V$ of $p$ such that $X\cap V$ is a submanifold. 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.

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