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real analytic subvariety
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(Definition)
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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 \begin{equation*} X \cap V = \{ p \in V \mid f(p) = 0 \text{ for all } f \in \mathcal{F} \}. \end{equation*}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.
- 1
- Jacek Bochnak, Michel Coste, Marie-Francoise Roy. Real Algebraic Geometry. Springer, 1998.
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"real analytic subvariety" is owned by jirka.
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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 9 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 2916 times total.
Classification:
| AMS MSC: | 14P05 (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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Pending Errata and Addenda
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