Incidence bounds on multijoints and generic joints

TL;DR

This paper establishes new bounds on the number of multijoints and generic joints formed by lines and algebraic curves in n-dimensional spaces over various fields, extending classical incidence results.

Contribution

It introduces bounds for multijoints and generic joints in higher dimensions and over different fields, generalizing previous results in real three-dimensional space.

Findings

Bound of (n-1) for multijoints in a^n with collections a_1,...,a_n

Bound of a(n-1) for generic joints in a^n with a lines, each in a lines

Extension of results to joints formed by real algebraic curves

Abstract

A point $x \in \mathbb{F}^n$ is a joint formed by a finite collection $\mathfrak{L}$ of lines in $\mathbb{F}^n$ if there exist at least $n$ lines in $\mathfrak{L}$ through $x$ that span $\mathbb{F}^n$. It is known that there are $\lesssim_n |\mathfrak{L}|^{\frac{n}{n-1}}$ joints formed by $\mathfrak{L}$. We say that a point $x \in \mathbb{F}^n$ is a multijoint formed by the finite collections $\mathfrak{L}_1,\ldots,\mathfrak{L}_n$ of lines in $\mathbb{F}^n$ if there exist at least $n$ lines through $x$, one from each collection, spanning $\mathbb{F}^n$. We show that there are $\lesssim_n (|\mathfrak{L}_1|\cdots |\mathfrak{L}_n|)^{\frac{1}{n-1}}$ such points for any field $\mathbb{F}$ and $n=3$, as well as for $\mathbb{F}=\mathbb{R}$ and any $n \geq 3$. Moreover, we say that a point $x \in \mathbb{F}^n$ is a generic joint formed by a finite collection $\mathfrak{L}$ of lines in $\mathbb{F}^n$ if each $n$ lines of $\mathfrak{L}$ through $x$ form a joint there. We show that, for $\mathbb{F}=\mathbb{R}$ and any $n \geq 3$, there are $\lesssim_n \frac{|\mathfrak{L}|^{\frac{n}{n-1}}}{k^{\frac{n+1}{n-1}}}+\frac{|\mathfrak{L}|}{k}$ generic joints formed by $\mathfrak{L}$, each lying in $\sim k$ lines of $\mathfrak{L}$. This result generalises, to all dimensions, a (very small) part of the main point-line incidence theorem in $\mathbb{R}^3$ in \cite{Guth_Katz_2010} by Guth and Katz. Finally, we generalise our results in $\mathbb{R}^n$ to the case of multijoints and generic joints formed by real algebraic curves.