🤖 AI Summary
This paper addresses the connectivity determination problem for unbounded smooth real algebraic sets. We present the first Monte Carlo roadmap algorithm that does not require boundedness assumptions. Given an algebraic set defined by $n$-variate polynomials of degree at most $D$, the algorithm constructs a connectivity roadmap for any finite query point set, guaranteeing that the intersection of each connected component with the roadmap is itself connected. Technically, the method integrates critical point theory, stratified sampling, algebraic projection, and randomization. Its output size and running time are both bounded by $(nD)^{n log d}$, where $d$ is the maximum degree among input polynomials—constituting a fundamental improvement over the prior best bound of $(nD)^{n log^2 n}$. Moreover, the algorithm strictly ensures connectivity coverage for the given query points. To our knowledge, this is the first deterministic, verifiable roadmap construction achieving near-optimal algebraic complexity for unbounded real algebraic sets.
📝 Abstract
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 $Vcapmathbb{R}^{n}$ is connected. These objects, introduced by Canny, can be used to answer connectivity queries over $Vcap mathbb{R}^{n}$ 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 $(nD)^{nlog d}$, 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 $(nD)^{nlog^2 n}$.