Foundations of Computational Mathematics

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

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

  • S. Basu
  • M.-F. Roy
  • M. Safey El Din
  • É. Schost


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 \(d^{O(k^{2})}.\)


Roadmaps Real algebraic variety Baby step-giant step 

Mathematics Subject Classification

Primary 14Q20 Secondary 14P05 68W05 



We are very grateful to the anonymous referees of the paper for their numerous suggestions. We are particularly grateful to one of them for pointing out an error in a preliminary version. The first author was supported in part by National Science Foundation Grants CCF-0915954, CCF-1319080, and DMS-1161629. The first and second authors did part of the work during a research stay in Oberwolfach as part of the Research in Pairs Programme. The third author is a member of Institut Universitaire de France and supported by a French National Research Agency EXACTA grant (ANR-09-BLAN-0371-01) and a GeoLMI grant (ANR-2011-BS03-011-06). The fourth author was supported by an NSERC Discovery Grant and by the Canada Research Chairs Program.


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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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