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real analytic subvariety (Definition)

Let $ U \subset {\mathbb{R}}^N$ be an open set.

Definition 1   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
$\displaystyle X \cap V = \{ p \in V \mid f(p) = 0$    for all $\displaystyle f \in \mathcal{F} \}.$    

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 2   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.

Bibliography

1
Jacek Bochnak, Michel Coste, Marie-Francoise Roy. Real Algebraic Geometry. Springer, 1998.



"real analytic subvariety" is owned by jirka.
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See Also: smooth submanifold contained in a subvariety of same dimension is real analytic

Other names:  real analytic variety, real analytic set
Also defines:  real algebraic variety, real algebraic subvariety, local real analytic subvariety, regular point, singular point

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Cross-references: semianalytic, complex, subvariety, submanifold, neighbourhood, regular, closed, polynomials, real, real analytic functions, point, closed set, open set
There are 8 references to this entry.

This is version 1 of real analytic subvariety, born on 2007-12-12.
Object id is 10125, canonical name is RealAnalyticSubvariety.
Accessed 1298 times total.

Classification:
AMS MSC14P05 (Algebraic geometry :: Real algebraic and real analytic geometry :: Real algebraic sets)
 14P15 (Algebraic geometry :: Real algebraic and real analytic geometry :: Real analytic and semianalytic sets)

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