algebraic manifold

Definition.

Let $k$ be a field and let $M\subset k^{n}$ be a submanifold. $M$ is said to be an algebraic manifold (or $k$ -algebraic) if there exists an irreducible algebraic variety $V\subset k^{n}$ such that $\dim V=\dim M$ and $M\subset V$ . If $k=\mathbb{R}$ , then $M$ is called a Nash manifold.

It can be proved that such a manifold is defined as the zero set of a finite collection of analytic algebraic functions.

References

1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.

Title	algebraic manifold
Canonical name	AlgebraicManifold
Date of creation	2013-03-22 15:36:08
Last modified on	2013-03-22 15:36:08
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	6
Author	jirka (4157)
Entry type	Definition
Classification	msc 14P20
Classification	msc 14-00
Classification	msc 58A07
Synonym	algebraic submanifold
Synonym	$k$ -algebraic manifold
Synonym	$k$ -algebraic submanifold
Defines	Nash manifold
Defines	Nash submanifold