Cutting Algebraic Curves into Pseudo-segments and Applications

TL;DR

This paper introduces a method to cut algebraic plane curves into pseudo-segments with at most one intersection per pair, leading to improved incidence bounds and complexity results for arrangements of algebraic curves.

Contribution

It extends pseudo-segment cutting techniques to algebraic curves of bounded degree, providing new bounds on incidences and arrangement complexities.

Findings

Cuts algebraic curves into pseudo-segments with near-optimal bounds.

Derives improved incidence bounds for points and algebraic curves.

Provides new complexity bounds for arrangements of algebraic curves.

Abstract

We show that a set of $n$ algebraic plane curves of constant maximum degree can be cut into $O(n^{3/2}\operatorname{polylog} n)$ Jordan arcs, so that each pair of arcs intersect at most once, i.e., they form a collection of pseudo-segments. This extends a similar (and slightly better) bound for pseudo-circles due to Marcus and Tardos. Our result is based on a technique of Ellenberg, Solymosi and Zahl that transforms arrangements of plane curves into arrangements of space curves, so that lenses (pairs of subarcs of the curves that intersect at least twice) become vertical depth cycles. We then apply a variant of a technique of Aronov and Sharir to eliminate these depth cycles by making a small number of cuts, which corresponds to a small number of cuts to the original planar arrangement of curves. After these cuts have been performed, the resulting curves form a collection of pseudo-segments. Our cutting bound leads to new incidence bounds between points and constant-degree algebraic curves. The conditions for these incidence bounds are slightly stricter than those for the current best-known bound of Pach and Sharir; for our result to hold, the curves must be algebraic and of bounded maximum degree, while Pach and Sharir's bound only imposes weaker, purely topological constraints on the curves. However, when our conditions hold, the new bounds are superior for almost all ranges of parameters. We also obtain new bounds on the complexity of a single level in an arrangement of constant-degree algebraic curves, and a new bound on the complexity of many marked faces in an arrangement of such curves.