Constructive Polynomial Partitioning for Algebraic Curves in $\mathbb{R}^3$ with Applications

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 ${\mathbb R}^d$ and for any positive integer $D$, there exists a polynomial of degree at most $D$ whose zero set divides ${\mathbb R}^d$ 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 ${{\mathbb R}}^3$. 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 $O(D^3\log^3{D})$ 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 $O(n^{4/3} { polylog }{n})$. 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 $O(n^{3/2+\varepsilon})$ pieces that are depth-cycle free for any $\varepsilon > 0$. The algorithm runs in $O(n^{3/2+\varepsilon})$ time, which is a considerable improvement over the previously known algorithms.