Generalized polar varieties: Geometry and algorithms
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 motiva...
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2005
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0885064X_v21_n4_p377_Bank http://hdl.handle.net/20.500.12110/paper_0885064X_v21_n4_p377_Bank |
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paper:paper_0885064X_v21_n4_p377_Bank2023-06-08T15:46:35Z Generalized polar varieties: Geometry and algorithms Arithmetic circuit Arithmetic network Complexity Elimination procedure Geometric degree Geometry of polar varieties and its generalizations Real polynomial equation solving Algorithms Computational complexity Digital arithmetic Matrix algebra Polynomials Probability Theorem proving Vectors Arithmetic circuit Arithmetic network Elimination procedure Geometric degree Real polynomial equation solving Computational geometry 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. © 2005 Elsevier Inc. All rights reserved. 2005 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0885064X_v21_n4_p377_Bank http://hdl.handle.net/20.500.12110/paper_0885064X_v21_n4_p377_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 |
Arithmetic circuit Arithmetic network Complexity Elimination procedure Geometric degree Geometry of polar varieties and its generalizations Real polynomial equation solving Algorithms Computational complexity Digital arithmetic Matrix algebra Polynomials Probability Theorem proving Vectors Arithmetic circuit Arithmetic network Elimination procedure Geometric degree Real polynomial equation solving Computational geometry |
| spellingShingle |
Arithmetic circuit Arithmetic network Complexity Elimination procedure Geometric degree Geometry of polar varieties and its generalizations Real polynomial equation solving Algorithms Computational complexity Digital arithmetic Matrix algebra Polynomials Probability Theorem proving Vectors Arithmetic circuit Arithmetic network Elimination procedure Geometric degree Real polynomial equation solving Computational geometry Generalized polar varieties: Geometry and algorithms |
| topic_facet |
Arithmetic circuit Arithmetic network Complexity Elimination procedure Geometric degree Geometry of polar varieties and its generalizations Real polynomial equation solving Algorithms Computational complexity Digital arithmetic Matrix algebra Polynomials Probability Theorem proving Vectors Arithmetic circuit Arithmetic network Elimination procedure Geometric degree Real polynomial equation solving Computational geometry |
| description |
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. © 2005 Elsevier Inc. All rights reserved. |
| title |
Generalized polar varieties: Geometry and algorithms |
| title_short |
Generalized polar varieties: Geometry and algorithms |
| title_full |
Generalized polar varieties: Geometry and algorithms |
| title_fullStr |
Generalized polar varieties: Geometry and algorithms |
| title_full_unstemmed |
Generalized polar varieties: Geometry and algorithms |
| title_sort |
generalized polar varieties: geometry and algorithms |
| publishDate |
2005 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0885064X_v21_n4_p377_Bank http://hdl.handle.net/20.500.12110/paper_0885064X_v21_n4_p377_Bank |
| _version_ |
1768542744739840000 |