Constructive polynomial partitioning for algebraic curves in R3 with applications

Boris Aronov, Esther Ezra, Joshua Zahl

    Research output: Contribution to journalArticlepeer-review

    Abstract

    In 2015, Guth [Math. Proc. Cambridge Philos. Soc., 159 (2015), pp. 459-469] proved that for any set of k-dimensional bounded complexity varieties in Rd and for any positive integer D, there exists a polynomial of degree at most D whose zero set divides Rd into open connected sets so that only a small fraction of the given varieties intersect each of these sets. Guth's result generalized an earlier result of Guth and Katz [Ann. Math., 181 (2015), pp. 155-190] 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 bounded-degree algebraic curves (or even lines) in R3. We present an efficient algorithmic construction for this setting. Given a set of n input algebraic curves and a positive integer D, we efficiently construct a decomposition of space into O(D3 log3 D) open “cells,” each of which meets O(n/D2) curves from the input. The construction time is O(n2). For the case of lines in 3-space, we present an improved implementation whose running time is O(n4/3 polylog n). The constant of proportionality in both time bounds depends on D and the maximum degree of the polynomials defining the input curves. As an application, we revisit the problem of eliminating depth cycles among nonvertical lines in 3-space, recently studied by Aronov and Sharir [Discrete Comput. Geom., 59 (2018), pp. 725-741] and show an algorithm that cuts n such lines 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 a considerable improvement over the previously known algorithms.

    Original languageEnglish (US)
    Pages (from-to)1109-1127
    Number of pages19
    JournalSIAM Journal on Computing
    Volume49
    Issue number6
    DOIs
    StatePublished - 2020

    Keywords

    • Algebraic methods in combinatorial geometry
    • Cycle elimination
    • Depth cycle
    • Depth order
    • Partitioning polynomial
    • ε-cutting

    ASJC Scopus subject areas

    • Computer Science(all)
    • Mathematics(all)

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