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.

翻译：回答实代数集中的连通性查询是有效实代数几何中的一个基本问题，在例如运动规划热点议题的机器人学等领域有许多应用。该计算问题通过计算所谓的道路图来解决，道路图是所研究集合V的实代数子集，维数至多为1，并且与V的所有半代数连通分量都有连通交集。计算道路图的算法依赖于对V的某些精心选取的子集建立连通性性质的结论，前提是假设V有界。在本文中，我们通过去掉V的有界性假设来扩展这些连通性结论。这利用了所谓广义极簇的性质，这些极簇是V关于某些精心选取的多项式映射的临界轨迹。