Computing roadmaps in unbounded smooth real algebraic sets II: algorithm and complexity
Rémi Prébet, Mohab Safey El Din, Éric Schost
TL;DR
This work extends roadmap computation to unbounded smooth real algebraic sets by introducing a Monte Carlo framework that bypasses the prior boundedness requirement. It leverages generalized polar varieties and incidence-based Lagrange systems to reorganize connectivity arguments, reducing the unbounded problem to bounded subproblems via finitely many fibers and a one-dimensional polar piece. The algorithm achieves complexity nearly matching the bounded-case state of the art, with careful control over the output size and probabilistic success via uniformly random parameter choices. The theoretical contributions include adapted Noether normalization, finiteness of fibers on generalized polar varieties, and atlas constructions for polar varieties and fibers, all integrated into a RoadmapBounded procedure that yields roadmaps for $(V,\mathcal{P})$. Practically, this enables robust connectivity queries and semi-algebraic component descriptions for unbounded smooth real algebraic sets defined over $\mathbb{Q}$, broadening applicability in robotics and related fields.
Abstract
A roadmap for an algebraic set $V$ defined by polynomials with coefficients in the field $\mathbb{Q}$ of rational numbers is an algebraic curve contained in $V$ whose intersection with all connected components of $V\cap\mathbb{R}^{n}$ is connected. These objects, introduced by Canny, can be used to answer connectivity queries over $V\cap \mathbb{R}^{n}$ provided that they are required to contain the finite set of query points $\mathcal{P}\subset V$; in this case, we say that the roadmap is associated to $(V, \mathcal{P})$. In this paper, we make effective a connectivity result we previously proved, to design a Monte Carlo algorithm which, on input (i) a finite sequence of polynomials defining $V$ (and satisfying some regularity assumptions) and (ii) an algebraic representation of finitely many query points $\mathcal{P}$ in $V$, computes a roadmap for $(V, \mathcal{P})$. This algorithm generalizes the nearly optimal one introduced by the last two authors by dropping a boundedness assumption on the real trace of $V$. The output size and running times of our algorithm are both polynomial in $(nD)^{n\log d}$, where $D$ is the maximal degree of the input equations and $d$ is the dimension of $V$. As far as we know, the best previously known algorithm dealing with such sets has an output size and running time respectively polynomial in $(n^{\log{n}}D)^{n\log n}$ and $(n^{\log{n}}D)^{n\log^2 n}$.