Computing roadmaps in unbounded smooth real algebraic sets II: algorithm and complexity
<|reference_start|>Computing roadmaps in unbounded smooth real algebraic sets II: algorithm and complexity: A roadmap for an algebraic set $V$ defined by polynomials with coefficients in some real field, say $\mathbb{R}$, 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 polynomial in $(nD)^{n\log^2 n}$.<|reference_end|>
@article{prébet2024computing,
    title={Computing roadmaps in unbounded smooth real algebraic sets II: algorithm
  and complexity},
    author={R'emi Pr'ebet and Mohab Safey El Din and 'Eric Schost},
    journal={arXiv preprint arXiv:2402.03111},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.03111},
    primaryClass={cs.SC math.AG}
}