Name
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 Q-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. |
Year | DOI | Venue |
|---|---|---|
2005 | 10.1016/j.jco.2004.10.001 | J. Complexity |
Keywords | DocType | Volume |
Arithmetic network,real trace,14P05,Arithmetic circuit,14B05,68W30,algorithmic elimination theory,68Q25,polar variety,Geometric degree,variety V,real number,Generalized polar variety,Complexity,generalized polar variety,Real polynomial equation solving,ambient space,classic notion,Elimination procedure,Geometry of polar varieties and its generalizations,complete intersection ideal,algebraic sample point | Journal | 21 |
Issue | ISSN | Citations |
4 | Journal of Complexity | 33 |
PageRank | References | Authors |
1.21 | 21 | 4 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| B. Bank | 1 | 98 | 7.45 |
| M. Giusti | 2 | 54 | 2.46 |
| J. Heintz | 3 | 162 | 19.20 |
| L. M. Pardo | 4 | 72 | 6.82 |