Constructive polynomial partitioning for algebraic curves in R3with applications

Boris Aronov, Esther Ezra, Joshua Zahl

    Research output: Contribution to conferencePaper

    Abstract

    In 2015, Guth proved that, for any set of k-dimensional varieties in Rdand for any positive integer D, there exists a polynomial of degree at most D whose zero-set divides Rdinto open connected “cells,” so that only a small fraction of the given varieties intersect each cell. Guth’s result generalized an earlier result of Guth and Katz for points. Guth’s proof relies on a variant of the Borsuk-Ulam theorem, and for k > 0, it is unknown how to obtain an explicit representation of such a partitioning polynomial and how to construct it efficiently. In particular, it is unknown how to effectively construct such a polynomial for curves (or even lines) in R3. We present an efficient algorithmic construction for this setting. Given a set of n input curves and a positive integer D, we efficiently construct a decomposition of space into O(D3log3D) open cells, each of which meets at most O(n/D2) curves from the input. The construction time is O(n2), where the constant of proportionality depends on D and the maximum degree of the polynomials defining the input curves. For the case of lines in 3-space we present an improved implementation, whose running time is O(n4/3polylog n). As an application, we revisit the problem of eliminating depth cycles among non-vertical pairwise disjoint triangles in 3-space, recently studied by Aronov et al. (2017) and De Berg (2017). Our main result is an algorithm that cuts n triangles into O(n3/2+ε) pieces that are depth cycle free, for any ε > 0. The algorithm runs in O(n3/2+ε) time, which is nearly worst-case optimal. We also sketch several other applications of our effective partitioning for curves in R3

    Original languageEnglish (US)
    Pages2636-2648
    Number of pages13
    StatePublished - 2019
    Event30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019 - San Diego, United States
    Duration: Jan 6 2019Jan 9 2019

    Conference

    Conference30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019
    CountryUnited States
    CitySan Diego
    Period1/6/191/9/19

    ASJC Scopus subject areas

    • Software
    • Mathematics(all)

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