# 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 |