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Conference Papers Year : 2023

Algorithm for connectivity queries on real algebraic curves

Abstract

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.
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Dates and versions

hal-04000614 , version 1 (22-02-2023)
hal-04000614 , version 2 (16-06-2023)

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Nazrul Islam, Adrien Poteaux, Rémi Prébet. Algorithm for connectivity queries on real algebraic curves. ISSAC 2023 - 48th International Symposium on Symbolic and Algebraic Computation, Jul 2023, Tromsø, Norway. ⟨10.1145/3597066.3597081⟩. ⟨hal-04000614v2⟩
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