A Baby Step–Giant Step Roadmap Algorithm for General Algebraic Sets
Identifieur interne : 000011 ( Hal/Curation ); précédent : 000010; suivant : 000012A Baby Step–Giant Step Roadmap Algorithm for General Algebraic Sets
Auteurs : Saugata Basu [États-Unis] ; Marie-Françoise Roy [France] ; Mohab Safey El Din [France] ; Eric Schost [Canada]Source :
- Foundations of Computational Mathematics [ 1615-3375 ] ; 2014.
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Abstract
Let R be a real closed field and D ⊂ R an ordered domain. We give an algorithm that takes as input a polynomial Q ∈ D[X 1 , . . . , X k ], and computes a description of a roadmap of the set of zeros, Zer(Q, R k), of Q in R k . The complexity of the algorithm, measured by the number of arithmetic opera-tions in the ordered domain D, is bounded by d O(k √ k) , where d = deg(Q) ≥ 2. As a consequence, there exist algorithms for computing the number of semi-algebraically connected components of a real algebraic set, Zer(Q, R k), whose complexity is also bounded by d O(k √ k) , where d = deg(Q) ≥ 2. The best pre-viously known algorithm for constructing a roadmap of a real algebraic subset of R k defined by a polynomial of degree d has complexity d O(k 2) .
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UWO</orgName><orgName type="acronym">London (CA)</orgName><desc><address><addrLine>1151 Richmond St, London, ON N6A 3K7, Canada</addrLine><country key="CA"></country></address></desc></org></tutelle></tutelles></hal:affiliation><country>Canada</country></affiliation></author></analytic><series><title level="j">Foundations of Computational Mathematics</title><idno type="ISSN">1615-3375</idno><imprint><date type="datePub">2014</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass><keywords scheme="mix" xml:lang="en"><term>Baby step-giant step</term><term>Real algebraic variety</term><term>Roadmaps</term></keywords></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Let R be a real closed field and D ⊂ R an ordered domain. We give an algorithm that takes as input a polynomial Q ∈ D[X 1 , . . . , X k ], and computes a description of a roadmap of the set of zeros, Zer(Q, R k), of Q in R k . The complexity of the algorithm, measured by the number of arithmetic opera-tions in the ordered domain D, is bounded by d O(k √ k) , where d = deg(Q) ≥ 2. As a consequence, there exist algorithms for computing the number of semi-algebraically connected components of a real algebraic set, Zer(Q, R k), whose complexity is also bounded by d O(k √ k) , where d = deg(Q) ≥ 2. The best pre-viously known algorithm for constructing a roadmap of a real algebraic subset of R k defined by a polynomial of degree d has complexity d O(k 2) .</div></front></TEI><hal api="V3"><titleStmt><title xml:lang="en">A Baby Step–Giant Step Roadmap Algorithm for General Algebraic Sets</title><author role="aut"><persName><forename type="first">Saugata</forename><surname>Basu</surname></persName><email></email><idno type="halauthor">387217</idno><affiliation ref="#struct-147250"></affiliation></author><author role="aut"><persName><forename type="first">Marie-Françoise</forename><surname>Roy</surname></persName><email></email><idno type="halauthor">387218</idno><affiliation ref="#struct-75"></affiliation></author><author role="aut"><persName><forename type="first">Mohab</forename><surname>Safey El Din</surname></persName><email></email><idno type="halauthor">1119202</idno><affiliation ref="#struct-179805"></affiliation></author><author role="aut"><persName><forename type="first">Eric</forename><surname>Schost</surname></persName><email></email><idno type="halauthor">66004</idno><affiliation ref="#struct-394846"></affiliation></author><editor role="depositor"><persName><forename>Mohab</forename><surname>Safey El Din</surname></persName><email>Mohab.Safey@lip6.fr</email></editor><funder ref="#projanr-37443"></funder></titleStmt><editionStmt><edition n="v1" type="current"><date type="whenSubmitted">2014-12-17 07:45:23</date><date type="whenWritten">2012-01-31</date><date type="whenModified">2015-11-24 16:30:25</date><date type="whenReleased">2014-12-19 08:51:03</date><date type="whenProduced">2014</date><date type="whenEndEmbargoed">2014-12-17</date><ref type="file" target="https://hal.inria.fr/hal-01096209/document"><date notBefore="2014-12-17"></date></ref><ref type="file" subtype="author" n="1" target="https://hal.inria.fr/hal-01096209/file/brss_revised-may-15-2014.pdf"><date notBefore="2014-12-17"></date></ref></edition><respStmt><resp>contributor</resp><name key="113611"><persName><forename>Mohab</forename><surname>Safey El Din</surname></persName><email>Mohab.Safey@lip6.fr</email></name></respStmt></editionStmt><publicationStmt><distributor>CCSD</distributor><idno type="halId">hal-01096209</idno><idno type="halUri">https://hal.inria.fr/hal-01096209</idno><idno type="halBibtex">basu:hal-01096209</idno><idno type="halRefHtml">Foundations of Computational Mathematics, Springer Verlag, 2014, 14 (6), pp.1117 - 1172. <10.1007/s10208-014-9212-1></idno><idno type="halRef">Foundations of Computational Mathematics, Springer Verlag, 2014, 14 (6), pp.1117 - 1172. <10.1007/s10208-014-9212-1></idno></publicationStmt><seriesStmt><idno type="stamp" n="UPMC">Université Pierre et Marie Curie</idno><idno type="stamp" n="UNIV-RENNES1">Université de Rennes 1</idno><idno type="stamp" n="IRMAR">Institut de Recherche Mathématique de Rennes</idno><idno type="stamp" n="CNRS">CNRS - Centre national de la recherche scientifique</idno><idno type="stamp" n="INRIA">INRIA - Institut National de Recherche en Informatique et en Automatique</idno><idno type="stamp" n="LIP6" p="UPMC">Laboratoire d'Informatique de Paris 6</idno><idno type="stamp" n="CHL">Centre Henri Lebesgue</idno><idno type="stamp" n="IRMAR-GAR" p="IRMAR">Géométrie algébrique réelle</idno><idno type="stamp" n="INRIA_TEST">INRIA - Institut National de Recherche en Informatique et en Automatique</idno><idno type="stamp" n="IRMAR-GAE" p="IRMAR">Géométrie et algèbre effectives </idno><idno type="stamp" n="UR2-HB">Université Rennes 2 - Haute Bretagne</idno><idno type="stamp" n="INRIA2">INRIA 2</idno></seriesStmt><notesStmt><note type="audience" n="2">International</note><note type="popular" n="0">No</note><note type="peer" n="1">Yes</note></notesStmt><sourceDesc><biblStruct><analytic><title xml:lang="en">A Baby Step–Giant Step Roadmap Algorithm for General Algebraic Sets</title><author role="aut"><persName><forename type="first">Saugata</forename><surname>Basu</surname></persName><idno type="halAuthorId">387217</idno><affiliation ref="#struct-147250"></affiliation></author><author role="aut"><persName><forename type="first">Marie-Françoise</forename><surname>Roy</surname></persName><idno type="halAuthorId">387218</idno><affiliation ref="#struct-75"></affiliation></author><author role="aut"><persName><forename type="first">Mohab</forename><surname>Safey El Din</surname></persName><idno type="halAuthorId">1119202</idno><affiliation ref="#struct-179805"></affiliation></author><author role="aut"><persName><forename type="first">Eric</forename><surname>Schost</surname></persName><idno type="halAuthorId">66004</idno><affiliation ref="#struct-394846"></affiliation></author></analytic><monogr><idno type="halJournalId" status="VALID">13455</idno><idno type="issn">1615-3375</idno><idno type="eissn">1615-3383</idno><title level="j">Foundations of Computational Mathematics</title><imprint><publisher>Springer Verlag</publisher><biblScope unit="volume">14</biblScope><biblScope unit="issue">6</biblScope><biblScope unit="pp">1117 - 1172</biblScope><date type="datePub">2014</date></imprint></monogr><idno type="doi">10.1007/s10208-014-9212-1</idno></biblStruct></sourceDesc><profileDesc><langUsage><language ident="en">English</language></langUsage><textClass><keywords scheme="author"><term xml:lang="en">Roadmaps</term><term xml:lang="en">Real algebraic variety</term><term xml:lang="en">Baby step-giant step</term></keywords><classCode scheme="acm" n="F.2"></classCode><classCode scheme="halDomain" n="info.info-sc">Computer Science [cs]/Symbolic Computation [cs.SC]</classCode><classCode scheme="halTypology" n="ART">Journal articles</classCode></textClass><abstract xml:lang="en">Let R be a real closed field and D ⊂ R an ordered domain. We give an algorithm that takes as input a polynomial Q ∈ D[X 1 , . . . , X k ], and computes a description of a roadmap of the set of zeros, Zer(Q, R k), of Q in R k . The complexity of the algorithm, measured by the number of arithmetic opera-tions in the ordered domain D, is bounded by d O(k √ k) , where d = deg(Q) ≥ 2. As a consequence, there exist algorithms for computing the number of semi-algebraically connected components of a real algebraic set, Zer(Q, R k), whose complexity is also bounded by d O(k √ k) , where d = deg(Q) ≥ 2. The best pre-viously known algorithm for constructing a roadmap of a real algebraic subset of R k defined by a polynomial of degree d has complexity d O(k 2) .</abstract></profileDesc></hal></record>Pour manipuler ce document sous Unix (Dilib)
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