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

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## 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## Notes

### Acknowledgments

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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