Constructive Polynomial Partitioning for Algebraic Curves in R^3 with Applications
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In 2015, Guth proved that for any set of k-dimensional varieties in R^d and for any positive integer D, there exists a polynomial of degree at most D whose zero-set divides R^d into 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 R^3. 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(D^3log^3D) open cells, each of which meets O(n/D^2) curves from the input. The construction time is O(n^2). For the case of lines in 3-space we present an improved implementation, whose running time is O(n^4/3polylog 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 non-vertical lines in 3-space, recently studied by Aronov and Sharir (2018), and show an algorithm that cuts n such lines into O(n^3/2+ε) pieces that are depth-cycle free, for any ε > 0. The algorithm runs in O(n^3/2+ε) time, which is a considerable improvement over previously known algorithms.
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