Foundations of Computational Mathematics

, Volume 14, Issue 6, pp 1117–1172 | Cite as

A Baby Step–Giant Step Roadmap Algorithm for General Algebraic Sets

Article

Abstract

Let \(\mathrm {R}\) be a real closed field and \(\mathrm{D}\subset \mathrm {R}\) an ordered domain. We present an algorithm that takes as input a polynomial \(Q \in \mathrm{D}[X_{1},\ldots ,X_{k}]\) and computes a description of a roadmap of the set of zeros, \(\mathrm{Zer}(Q,\,\mathrm {R}^{k}),\) of Q in \(\mathrm {R}^{k}.\) The complexity of the algorithm, measured by the number of arithmetic operations in the ordered domain \(\mathrm{D},\) is bounded by \(d^{O(k \sqrt{k})},\) where \(d = \deg (Q)\ge 2.\) As a consequence, there exist algorithms for computing the number of semialgebraically connected components of a real algebraic set, \(\mathrm{Zer}(Q,\,\mathrm {R}^{k}),\) whose complexity is also bounded by \(d^{O(k \sqrt{k})},\) where \(d = \deg (Q)\ge 2.\) The best previously known algorithm for constructing a roadmap of a real algebraic subset of \(\mathrm {R}^{k}\) defined by a polynomial of degree d has complexity \(d^{O(k^{2})}.\)

Keywords

Roadmaps Real algebraic variety Baby step-giant step 

Mathematics Subject Classification

Primary 14Q20 Secondary 14P05 68W05 

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Copyright information

© SFoCM 2014

Authors and Affiliations

  • S. Basu
    • 1
  • M.-F. Roy
    • 2
  • M. Safey El Din
    • 3
  • É. Schost
    • 4
  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA
  2. 2.IRMAR (URA CNRS 305)Université de Rennes 1Rennes CedexFrance
  3. 3.Sorbonne Universités, UPMC, Univ. Paris 06, UMR CNRS 7606, LIP6, INRIA Paris-Rocquencourt Center PolSys ProjectInstitut Universitaire de FranceParisFrance
  4. 4.Computer Science DepartmentThe University of Western OntarioLondonCanada

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