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 \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