Back to Search
Start Over
Generalized polar varieties: geometry and algorithms
- Source :
-
Journal of Complexity . Aug2005, Vol. 21 Issue 4, p377-412. 36p. - Publication Year :
- 2005
-
Abstract
- Abstract: Let V be a closed algebraic subvariety of the n-dimensional projective space over the complex or real numbers and suppose that V is non-empty and equidimensional. The classic notion of a polar variety of V associated with a given linear subvariety of the ambient space of V was generalized and motivated in Bank et al. (Kybernetika 40 (2004), to appear). As particular instances of this notion of a generalized polar variety one reobtains the classic one and an alternative type of a polar variety, called dual. As main result of the present paper we show that for a generic choice of their parameters the generalized polar varieties of V are empty or equidimensional and smooth in any regular point of V. In the case that the variety V is affine and smooth and has a complete intersection ideal of definition, we are able, for a generic parameter choice, to describe locally the generalized polar varieties of V by explicit equations. Finally, we indicate how this description may be used in order to design in the context of algorithmic elimination theory a highly efficient, probabilistic elimination procedure for the following task: In case, that the variety V is -definable and affine, having a complete intersection ideal of definition, and that the real trace of V is non-empty and smooth, find for each connected component of the real trace of V an algebraic sample point. [Copyright &y& Elsevier]
- Subjects :
- *GEOMETRY
*ALGORITHMS
*ALGEBRA
*EUCLIDEAN algorithm
Subjects
Details
- Language :
- English
- ISSN :
- 0885064X
- Volume :
- 21
- Issue :
- 4
- Database :
- Academic Search Index
- Journal :
- Journal of Complexity
- Publication Type :
- Academic Journal
- Accession number :
- 18153687
- Full Text :
- https://doi.org/10.1016/j.jco.2004.10.001