real analytic subvariety

Let $U\subset{\mathbb{R}}^{N}$ 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 $\mathcal{F}$ of real analytic functions defined in $V$, such that

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

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

References

• 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
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