2003-06-24Buch DOI: 10.18452/2603
A first approach to generalized polar varieties
 dc.contributor.author Bank, Bernd dc.contributor.author Giusti, Marc dc.contributor.author Heintz, Joos dc.contributor.author Pardo, Luis-Miguel dc.date.accessioned 2017-06-15T17:41:42Z dc.date.available 2017-06-15T17:41:42Z dc.date.created 2005-11-03 dc.date.issued 2003-06-24 dc.identifier.issn 0863-0976 dc.identifier.uri http://edoc.hu-berlin.de/18452/3255 dc.description.abstract Let W be a closed algebraic subvariety of the n-dimensional projective space over the complex or real numbers and suppose that W is non--empty and equidimensional. In this paper we generalize the classic notion of polar variety of W associated with a given linear subvariety of the ambient space of W. As particular instances of this new notion of generalized polar variety we reobtain the classic ones and two new types of polar varieties, called {\it dual} and (in case that W is affine) {\it conic}. We show that for a generic choice of their parameters the generalized polar varieties of W are empty or equidimensional and, if W is smooth, that their ideals of definition are Cohen--Macaulay. In the case that the variety W 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 W 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 W is \Q-definable and affine, having a complete intersection ideal of definition, and that the real trace of W is non--empty and smooth, find for each connected component of the real trace of W a representative point. eng dc.language.iso eng dc.publisher Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik dc.rights.uri http://rightsstatements.org/vocab/InC/1.0/ dc.subject complexity eng dc.subject geometric degree eng dc.subject arithmetic network eng dc.subject Geometry of polar varieties and its generalizations eng dc.subject real polynomial equation solving eng dc.subject elimination procedure eng dc.subject arithmetic circuit eng dc.subject.ddc 510 Mathematik dc.title A first approach to generalized polar varieties dc.type book dc.identifier.urn urn:nbn:de:kobv:11-10052373 dc.identifier.doi http://dx.doi.org/10.18452/2603 local.edoc.container-title Preprints aus dem Institut für Mathematik local.edoc.pages 39 local.edoc.type-name Buch local.edoc.container-type series local.edoc.container-type-name Schriftenreihe local.edoc.container-volume 2003 local.edoc.container-issue 5 local.edoc.container-year 2003 local.edoc.container-erstkatid 2075199-0