Generalized polar varieties: Geometry and algorithms

• Autores: Bank, B.; Giusti, M.; Heintz, J.; Pardo, L.M.
• Año de publicación: 2005
• Idioma: inglés
• Tipo de recurso: artículo
• Fuente: J. Complexity 2005;21(4):377-412
• Materia: 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
• Institución: Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales

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.