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Preprints, Working Papers, ... Year : 2022

Generalized polar varieties, fibers and connectivity in unbounded smooth real algebraic sets

Abstract

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.
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Dates and versions

hal-03508000 , version 1 (03-01-2022)
hal-03508000 , version 2 (07-03-2022)
hal-03508000 , version 3 (07-06-2023)

Identifiers

  • HAL Id : hal-03508000 , version 1

Cite

Rémi Prébet, Mohab Safey El Din, Éric Schost. Generalized polar varieties, fibers and connectivity in unbounded smooth real algebraic sets. 2022. ⟨hal-03508000v1⟩
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