Properness defects of projections and computation of one point in each connected component of a real algebraic set
Identifieur interne : 003611 ( Crin/Corpus ); précédent : 003610; suivant : 003612Properness defects of projections and computation of one point in each connected component of a real algebraic set
Auteurs : Mohab Safey El Din ; Eric SchostSource :
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Abstract
Computing at least one point in each connected component of a real algebraic set is a basic subroutine to decide emptiness of semi-algbraic sets, which is a fundamental algorithmic problem in effective real algebraic geometry. In this article, we propose a new algorithm for this task, which avoids a hypothesis of properness required in many of the previous methods. We show how studying the set of non-properness of a linear projection enables to detect connected components of a real algebraic set without critical points. Our algorithm is based on this result and its practical counterpoint, using the triangular representation of algebraic varieties. Our experiments show its efficiency on a family of examples.
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<front><div type="abstract" xml:lang="en" wicri:score="2379">Computing at least one point in each connected component of a real algebraic set is a basic subroutine to decide emptiness of semi-algbraic sets, which is a fundamental algorithmic problem in effective real algebraic geometry. In this article, we propose a new algorithm for this task, which avoids a hypothesis of properness required in many of the previous methods. We show how studying the set of non-properness of a linear projection enables to detect connected components of a real algebraic set without critical points. Our algorithm is based on this result and its practical counterpoint, using the triangular representation of algebraic varieties. Our experiments show its efficiency on a family of examples.</div>
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<BibTex type="techreport"><ref>safey_el_din02a</ref>
<crinnumber>A02-R-465</crinnumber>
<category>15</category>
<equipe>SPACES</equipe>
<author><e>Safey El Din, Mohab</e>
<e>Schost, Eric</e>
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<title>Properness defects of projections and computation of one point in each connected component of a real algebraic set</title>
<institution>INRIA</institution>
<year>2002</year>
<type>Rapport de recherche</type>
<month>Oct</month>
<url>http://www.loria.fr/publications/2002/A02-R-465/A02-R-465.ps</url>
<keywords><e>polynomial systems</e>
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<abstract>Computing at least one point in each connected component of a real algebraic set is a basic subroutine to decide emptiness of semi-algbraic sets, which is a fundamental algorithmic problem in effective real algebraic geometry. In this article, we propose a new algorithm for this task, which avoids a hypothesis of properness required in many of the previous methods. We show how studying the set of non-properness of a linear projection enables to detect connected components of a real algebraic set without critical points. Our algorithm is based on this result and its practical counterpoint, using the triangular representation of algebraic varieties. Our experiments show its efficiency on a family of examples.</abstract>
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