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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. 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.

Registro:

Documento: Artículo
Título:Generalized polar varieties and an efficient real elimination procedure
Autor:Bank, B.; Giusti, M.; Heintz, J.; Pardo, L.M.
Filiación:Humboldt-Universität zu Berlin, Institut Für Mathematik, 1009, Germany
Laboratoire STIX, École Polytechnique, 91228 Palaiseau Cedex, France
Departamento de Computación, Fac. de Ciencias Exactas y Naturales, Ciudad Univ, Pab.I, 1428 Buenos Aires, Argentina
CONICET, Argentina
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cantabria, 39071 Santander, Spain
Palabras clave: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
Año:2004
Volumen:40
Número:5
Página de inicio:519
Página de fin:550
Título revista:Kybernetika
Título revista abreviado:Kybernetika
ISSN:00235954
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00235954_v40_n5_p519_Bank

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Citas:

---------- APA ----------
Bank, B., Giusti, M., Heintz, J. & Pardo, L.M. (2004) . Generalized polar varieties and an efficient real elimination procedure. Kybernetika, 40(5), 519-550.
Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00235954_v40_n5_p519_Bank [ ]
---------- CHICAGO ----------
Bank, B., Giusti, M., Heintz, J., Pardo, L.M. "Generalized polar varieties and an efficient real elimination procedure" . Kybernetika 40, no. 5 (2004) : 519-550.
Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00235954_v40_n5_p519_Bank [ ]
---------- MLA ----------
Bank, B., Giusti, M., Heintz, J., Pardo, L.M. "Generalized polar varieties and an efficient real elimination procedure" . Kybernetika, vol. 40, no. 5, 2004, pp. 519-550.
Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00235954_v40_n5_p519_Bank [ ]
---------- VANCOUVER ----------
Bank, B., Giusti, M., Heintz, J., Pardo, L.M. Generalized polar varieties and an efficient real elimination procedure. Kybernetika. 2004;40(5):519-550.
Available from: https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00235954_v40_n5_p519_Bank [ ]