Computing roadmaps in unbounded smooth real algebraic sets I: Connectivity results

Answering connectivity queries in real algebraic sets is a fundamental problem in effective real algebraic geometry that finds many applications in e.g. robotics where motion planning issues are topical. This computational problem is tackled through the computation of so-called roadmaps which are re...

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Bibliographic Details
Published in:Journal of symbolic computation Vol. 120; p. 102234
Main Authors: Prébet, Rémi, Safey El Din, Mohab, Schost, Éric
Format: Journal Article
Language:English
Published: Elsevier Ltd 01.01.2024
Elsevier
Subjects:
Algebraic Geometry
Computational real algebraic geometry
Computer Science
Critical points
Mathematics
Real algebraic sets
Roadmaps
Symbolic Computation
14Q30
68W30
Critical points
14Q20
Real algebraic sets
14P05
Roadmaps
Computational real algebraic geometry
ISSN:0747-7171, 1095-855X
Online Access:Get full text
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Summary:Answering connectivity queries in real algebraic sets is a fundamental problem in effective real algebraic geometry that finds many applications in e.g. robotics where motion planning issues are topical. This computational problem is tackled through the computation of so-called roadmaps which are real algebraic subsets of the set V under study, of dimension at most one, and which have a connected intersection with all semi-algebraically connected components of V. Algorithms for computing roadmaps rely on statements establishing connectivity properties of some well-chosen subsets of V, assuming that V is bounded. In this paper, we extend such connectivity statements by dropping the boundedness assumption on V. This exploits properties of so-called generalized polar varieties, which are critical loci of V for some well-chosen polynomial maps.
ISSN:0747-7171
1095-855X
DOI:10.1016/j.jsc.2023.102234