On the geometry of polar varieties

We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are necessary to prove the correctness and complexity estimates of these...

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Bibliographic Details
Published in:Applicable algebra in engineering, communication and computing communication and computing, 2010, Vol.21 (1), p.33-83
Main Authors: Bank, Bernd, Giusti, Marc, Heintz, Joos, Safey El Din, Mohab, Schost, Eric
Format: Article
Language:English
Subjects:
Artificial Intelligence
Computer Hardware
Computer Science
Symbolic and Algebraic Manipulation
Theory of Computation
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Summary:We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are necessary to prove the correctness and complexity estimates of these algorithms. Our results form also the geometrical main ingredients for the computational treatment of singular hypersurfaces. In particular, we show the non–emptiness of suitable generic dual polar varieties of (possibly singular) real varieties, show that generic polar varieties may become singular at smooth points of the original variety and exhibit a sufficient criterion when this is not the case. Further, we introduce the new concept of meagerly generic polar varieties and give a degree estimate for them in terms of the degrees of generic polar varieties. The statements are illustrated by examples and a computer experiment.
ISSN:0938-1279
1432-0622
DOI:10.1007/s00200-009-0117-1