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

Rémi Prébet
• Function : Author
Mohab Safey El Din
• Function : Author
Éric Schost
• Function : Author

#### 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 \emph{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 \emph{generalized polar varieties}, which are critical loci of $V$ for some well-chosen polynomial maps.

### Dates and versions

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

### Identifiers

• HAL Id : hal-03508000 , version 2

### Cite

Rémi Prébet, Mohab Safey El Din, Éric Schost. Computing roadmaps in unbounded smooth real algebraic sets I: connectivity results. 2022. ⟨hal-03508000v2⟩

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