We consider the problem of answering connectivity queries on a real algebraic curve. The curve is given as the real trace of an algebraic curve, assumed to be in generic position, and being defined by some rational parametrizations. The query points are given by a zero-dimensional parametrization. We design an algorithm which counts the number of connected components of the real curve under study, and decides which query point lie in which connected component, in time log-linear in $N^6$, where $N$ is the maximum of the degrees and coefficient bit-sizes of the polynomials given as input. This matches the currently best-known bound for computing the topology of real plane curves. The main novelty of this algorithm is the avoidance of the computation of the complete topology of the curve.

翻译：我们考虑在实代数曲线上回答连通性查询的问题。该曲线给定为代数曲线的实迹，假设处于一般位置，并通过若干有理参数化定义。查询点由零维参数化给出。我们设计了一种算法，该算法计算所研究实曲线的连通分量数量，并判定每个查询点所属的连通分量，其时间复杂度为 $N^6$ 的对数线性阶，其中 $N$ 是输入多项式的最大次数与系数比特大小。该结果匹配当前已知的实平面曲线拓扑计算最优界。算法的主要创新在于避免了计算曲线的完整拓扑结构。