Generalized polar varieties and an efficient real elimination procedure
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. In this paper we generalize the classic notion of polar variety of V associated with a given linear subvariety of the ambient space of V....
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2004
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00235954_v40_n5_p519_Bank http://hdl.handle.net/20.500.12110/paper_00235954_v40_n5_p519_Bank |
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paper:paper_00235954_v40_n5_p519_Bank2023-06-08T14:51:28Z Generalized polar varieties and an efficient real elimination procedure Arithmetic circuit Arithmetic network Complexity Elimination procedure Geometric degree Geometry of polar varieties and its generalizations Real polynomial equation solving Arithmetic circuit Arithmetic networks Elimination procedure Geometric degree Geometry of polar varieties Real polynomial equation solving Algorithms Computational complexity Degrees of freedom (mechanics) General purpose computers Parameter estimation Polynomial approximation Problem solving Real time systems 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. In this paper we generalize the classic notion of polar variety of V associated with a given linear subvariety of the ambient space of V. As particular instances of this new notion of generalized polar variety we reobtain the classic ones and two new types of polar varieties, called dual and (in case that V is affine) conic. We show that for a generic choice of their parameters the generalized polar varieties of V are either empty or equidimensional and, if V is smooth, that their ideals of definition are Cohen-Macaulay. 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 use this description in order to design a new, highly efficient elimination procedure for the following algorithmic 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 a representative point. 2004 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00235954_v40_n5_p519_Bank http://hdl.handle.net/20.500.12110/paper_00235954_v40_n5_p519_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 Arithmetic circuit Arithmetic networks Elimination procedure Geometric degree Geometry of polar varieties Real polynomial equation solving Algorithms Computational complexity Degrees of freedom (mechanics) General purpose computers Parameter estimation Polynomial approximation Problem solving Real time systems Geometry |
| spellingShingle |
Arithmetic circuit Arithmetic network Complexity Elimination procedure Geometric degree Geometry of polar varieties and its generalizations Real polynomial equation solving Arithmetic circuit Arithmetic networks Elimination procedure Geometric degree Geometry of polar varieties Real polynomial equation solving Algorithms Computational complexity Degrees of freedom (mechanics) General purpose computers Parameter estimation Polynomial approximation Problem solving Real time systems Geometry Generalized polar varieties and an efficient real elimination procedure |
| topic_facet |
Arithmetic circuit Arithmetic network Complexity Elimination procedure Geometric degree Geometry of polar varieties and its generalizations Real polynomial equation solving Arithmetic circuit Arithmetic networks Elimination procedure Geometric degree Geometry of polar varieties Real polynomial equation solving Algorithms Computational complexity Degrees of freedom (mechanics) General purpose computers Parameter estimation Polynomial approximation Problem solving Real time systems 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. In this paper we generalize the classic notion of polar variety of V associated with a given linear subvariety of the ambient space of V. As particular instances of this new notion of generalized polar variety we reobtain the classic ones and two new types of polar varieties, called dual and (in case that V is affine) conic. We show that for a generic choice of their parameters the generalized polar varieties of V are either empty or equidimensional and, if V is smooth, that their ideals of definition are Cohen-Macaulay. 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 use this description in order to design a new, highly efficient elimination procedure for the following algorithmic 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 a representative point. |
| title |
Generalized polar varieties and an efficient real elimination procedure |
| title_short |
Generalized polar varieties and an efficient real elimination procedure |
| title_full |
Generalized polar varieties and an efficient real elimination procedure |
| title_fullStr |
Generalized polar varieties and an efficient real elimination procedure |
| title_full_unstemmed |
Generalized polar varieties and an efficient real elimination procedure |
| title_sort |
generalized polar varieties and an efficient real elimination procedure |
| publishDate |
2004 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00235954_v40_n5_p519_Bank http://hdl.handle.net/20.500.12110/paper_00235954_v40_n5_p519_Bank |
| _version_ |
1768541785867419648 |