On the geometry of polar varieties

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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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Autores principales:	Bank, B., Giusti, M., Heintz, J., Safey El Din, M., Schost, E.
Formato:	JOUR
Materias:	Real polynomial equation solving, Singularities, Classic polar varieties, Dual polar varieties, Generic polar varieties, Meagerly generic polar varieties Computational complexity Polynomials Algebraic varieties Complexity bounds Complexity estimates Computer experiment Hyper-surfaces Real polynomial equation solving Sufficient criterion Geometry
Acceso en línea:	http://hdl.handle.net/20.500.12110/paper_09381279_v21_n1_p33_Bank
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	todo:paper_09381279_v21_n1_p33_Bank
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spelling	todo:paper_09381279_v21_n1_p33_Bank2023-10-03T15:48:45Z On the geometry of polar varieties Bank, B. Giusti, M. Heintz, J. Safey El Din, M. Schost, E. Real polynomial equation solving, Singularities, Classic polar varieties, Dual polar varieties, Generic polar varieties, Meagerly generic polar varieties Computational complexity Polynomials Algebraic varieties Complexity bounds Complexity estimates Computer experiment Hyper-surfaces Real polynomial equation solving Sufficient criterion Geometry 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. © 2009 Springer-Verlag. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_09381279_v21_n1_p33_Bank
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	Real polynomial equation solving, Singularities, Classic polar varieties, Dual polar varieties, Generic polar varieties, Meagerly generic polar varieties Computational complexity Polynomials Algebraic varieties Complexity bounds Complexity estimates Computer experiment Hyper-surfaces Real polynomial equation solving Sufficient criterion Geometry
spellingShingle	Real polynomial equation solving, Singularities, Classic polar varieties, Dual polar varieties, Generic polar varieties, Meagerly generic polar varieties Computational complexity Polynomials Algebraic varieties Complexity bounds Complexity estimates Computer experiment Hyper-surfaces Real polynomial equation solving Sufficient criterion Geometry Bank, B. Giusti, M. Heintz, J. Safey El Din, M. Schost, E. On the geometry of polar varieties
topic_facet	Real polynomial equation solving, Singularities, Classic polar varieties, Dual polar varieties, Generic polar varieties, Meagerly generic polar varieties Computational complexity Polynomials Algebraic varieties Complexity bounds Complexity estimates Computer experiment Hyper-surfaces Real polynomial equation solving Sufficient criterion Geometry
description	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. © 2009 Springer-Verlag.
format	JOUR
author	Bank, B. Giusti, M. Heintz, J. Safey El Din, M. Schost, E.
author_facet	Bank, B. Giusti, M. Heintz, J. Safey El Din, M. Schost, E.
author_sort	Bank, B.
title	On the geometry of polar varieties
title_short	On the geometry of polar varieties
title_full	On the geometry of polar varieties
title_fullStr	On the geometry of polar varieties
title_full_unstemmed	On the geometry of polar varieties
title_sort	on the geometry of polar varieties
url	http://hdl.handle.net/20.500.12110/paper_09381279_v21_n1_p33_Bank
work_keys_str_mv	AT bankb onthegeometryofpolarvarieties AT giustim onthegeometryofpolarvarieties AT heintzj onthegeometryofpolarvarieties AT safeyeldinm onthegeometryofpolarvarieties AT schoste onthegeometryofpolarvarieties
_version_	1807315281859575808

On the geometry of polar varieties

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