Incidences with curves and surfaces in three dimensions, with applications to distinct and repeated distances

TL;DR

This paper advances the understanding of incidences between points and algebraic curves or surfaces in three-dimensional space, providing new bounds and applications to classic problems like distances and similar triangles.

Contribution

It refines algebraic geometry tools to derive improved incidence bounds for points with curves and surfaces in 3D, generalizing previous results and applying them to distance problems.

Findings

New bounds for point-curve incidences in 3D

Enhanced bounds for point-surface incidences in 3D

Improved estimates for distances and similar triangles in 3D

Abstract

We study a wide spectrum of incidence problems involving points and curves or points and surfaces in $\mathbb R^3$. The current (and in fact the only viable) approach to such problems, pioneered by Guth and Katz [2010,2015], requires a variety of tools from algebraic geometry, most notably (i) the polynomial partitioning technique, and (ii) the study of algebraic surfaces that are ruled by lines or, in more recent studies [Guth-Zahl 2016], by algebraic curves of some constant degree. By exploiting and refining these tools, we obtain new and improved bounds for numerous incidence problems in $\mathbb R^3$. In broad terms, we consider two kinds of problems, those involving points and constant-degree algebraic \emph{curves}, and those involving points and constant-degree algebraic \emph{surfaces}. In some variants we assume that the points lie on some fixed constant-degree algebraic variety, and in others we consider arbitrary sets of points in 3-space. our results provide a "grand generalization" of most of the previous studies of (special instances of) previous works for both curves and surfaces. As an application of our point-curve incidence bound, we consider the problem of bounding the number of similar triangles spanned by a set of $n$ points in $\mathbb R^3$, and obtain the bound $O(n^{15/7}),$ thus improving the bound of Agarwal et al [2007]. As applications of our point-surface incidence bounds, we consider the problems of distinct and repeated distances determined by a set of $n$ points in $\mathbb R^3$, two of the most celebrated open problems in combinatorial geometry. We obtain new and improved bounds for two special cases, one in which the points lie on some algebraic variety of constant degree, and one involving incidences between pairs in $P_1\times P_2$, where $P_1$ is contained in a variety and $P_2$ is arbitrary.