Joints tightened

TL;DR

This paper establishes a near-optimal upper bound on the number of joints formed by lines in d-dimensional space, matching known constructions and confirming Guth's conjecture using advanced polynomial methods.

Contribution

It introduces a new upper bound on joints in d-dimensional space that aligns with the best known constructions, extending polynomial techniques to multijoints.

Findings

Proved a new upper bound on the number of joints.

Matched the bound to the known optimal construction.

Extended polynomial method techniques to multijoints.

Abstract

In $d$-dimensional space (over any field), given a set of lines, a joint is a point passed through by $d$ lines not all lying in some hyperplane. The joints problem asks to determine the maximum number of joints formed by $L$ lines, and it was one of the successes of the Guth--Katz polynomial method. We prove a new upper bound on the number of joints that matches, up to a $1+o(1)$ factor, the best known construction: place $k$ generic hyperplanes, and use their $(d-1)$-wise intersections to form $\binom{k}{d-1}$ lines and their $d$-wise intersections to form $\binom{k}{d}$ joints. Guth conjectured that this construction is optimal. Our technique builds on the work on Ruixiang Zhang proving the multijoints conjecture via an extension of the polynomial method. We set up a variational problem to control the high order of vanishing of a polynomial at each joint.