Article Dans Une Revue Journal of Symbolic Computation Année : 2026

Computing roadmaps in unbounded smooth real algebraic sets II: algorithm and complexity

Résumé

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 polynomial in $(nD)^{n\log^2 n}$. 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 $(n^{\log{n}}D)^{n\log^2 n}$.

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Dates et versions

hal-04439518 , version 1 (05-02-2024)
hal-04439518 , version 2 (19-11-2025)

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Rémi Prébet, Mohab Safey El Din, Éric Schost. Computing roadmaps in unbounded smooth real algebraic sets II: algorithm and complexity. Journal of Symbolic Computation, inPress, 135, pp.102532. ⟨10.1016/j.jsc.2025.102532⟩. ⟨hal-04439518v2⟩
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