A Baby Step–Giant Step Roadmap Algorithm for General Algebraic Sets
Identifieur interne : 000314 ( Main/Exploration ); précédent : 000313; suivant : 000315A 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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Le document en format XML
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<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>
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<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>
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<country name="Canada"><noRegion><name sortKey="Schost, Eric" sort="Schost, Eric" uniqKey="Schost E" first="Eric" last="Schost">Eric Schost</name>
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