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

Rémi Prébet (1) , Mohab Safey El Din (1) , Éric Schost (2)

1 PolSys - Polynomial Systems
2 David R. Cheriton School of Computer Science

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.

Mots clés: Computational real algebraic geometry; Real algebraic sets; Critical points; Roadmaps

Domaines: Calcul formel [cs.SC]; Géométrie algébrique [math.AG]

Rémi Prébet, Mohab Safey El Din, Éric Schost. Computing roadmaps in unbounded smooth real algebraic sets I: connectivity results. Journal of Symbolic Computation, 2024, 120, pp.102234. ⟨10.1016/j.jsc.2023.102234⟩. ⟨hal-03508000v3⟩