A Baby Step–Giant Step Roadmap Algorithm for General Algebraic Sets
Identifieur interne : 000113 ( France/Analysis ); précédent : 000112; suivant : 000114A 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.
English descriptors
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) .
Url:
Affiliations:
- Canada, France, États-Unis
- Région Bretagne
- Rennes
- Université Rennes 2 Haute Bretagne, Université de Rennes 1, Université européenne de Bretagne
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Hal:hal-01096209Le document en format XML
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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><affiliations><list><country><li>Canada</li><li>France</li><li>États-Unis</li></country><region><li>Région Bretagne</li></region><settlement><li>Rennes</li></settlement><orgName><li>Université Rennes 2 Haute Bretagne</li><li>Université de Rennes 1</li><li>Université européenne de Bretagne</li></orgName></list><tree><country name="États-Unis"><noRegion><name sortKey="Basu, Saugata" sort="Basu, Saugata" uniqKey="Basu S" first="Saugata" last="Basu">Saugata Basu</name></noRegion></country><country name="France"><region name="Région Bretagne"><name sortKey="Roy, Marie Francoise" sort="Roy, Marie Francoise" uniqKey="Roy M" first="Marie-Françoise" last="Roy">Marie-Françoise Roy</name></region><name sortKey="Safey El Din, Mohab" sort="Safey El Din, Mohab" uniqKey="Safey El Din M" first="Mohab" last="Safey El Din">Mohab Safey El Din</name></country><country name="Canada"><noRegion><name sortKey="Schost, Eric" sort="Schost, Eric" uniqKey="Schost E" first="Eric" last="Schost">Eric Schost</name></noRegion></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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