real analytic subvarietyRealAnalyticSubvariety2007-12-12 16:56:372007-12-12 16:56:37Definitionreal algebraic varietyreal algebraic subvarietylocal real analytic subvarietyregular pointsingular point% this is the default PlanetMath preamble. as your knowledge
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Let $U \subset {\mathbb{R}}^N$ be an open set.
\begin{defn}
A closed set $X \subset U$ is called a {\em 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 {\em real algebraic subvariety}.
\end{defn}
If $X$ is not required to be closed, then it is said to be a {\em 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.
\begin{defn}
A point $p \in X$ is called a {\em regular point} if there is a neighbourhood
$V$ of $p$ such that $X \cap V$ is a submanifold. Any other
point is called a {\em singular point}.
\end{defn}
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.
\begin{thebibliography}{9}
\bibitem{Whitney:varieties}
Jacek Bochnak, Michel Coste, Marie-Francoise Roy.
{\em \PMlinkescapetext{Real Algebraic Geometry}}.
Springer, 1998.
\end{thebibliography}