Counting joints with multiplicities

TL;DR

This paper establishes an upper bound on the sum of square roots of joint multiplicities formed by lines in three-dimensional space, extending the result to algebraic curves and polynomial parametrizations.

Contribution

It introduces a bound on the sum of joint multiplicities' square roots for lines and extends this to algebraic curves and polynomial parametrizations in A3^3.

Findings

Sum of square roots of joint multiplicities is bounded by L^{3/2}

Extension of results to algebraic curves of bounded degree

Extension to polynomial-parametrized curves in A3^3

Abstract

Let $\mathfrak{L}$ be a collection of $L$ lines in $\R^3$ and $J$ the set of joints formed by $\mathfrak{L}$, i.e. the set of points each of which lies in at least 3 non-coplanar lines of $\mathfrak{L}$. It is known that $|J| \lesssim L^{3/2}$ (first proved by Guth and Katz). For each joint $x \in J$, let the multiplicity $N(x)$ of $x$ be the number of triples of non-coplanar lines through $x$. We prove here that $\sum_{x \in J}N(x)^{1/2} \lesssim L^{3/2}$, while in the last section we extend this result to real algebraic curves of uniformly bounded degree in $\R^3$, as well as to curves in $\R^3$ parametrised by real polynomials of uniformly bounded degree.