A proof of the Multijoints Conjecture and Carbery's generalization

Abstract

Given a set of lines in a $d$-dimensional linear space ${\mathbb{F}}^d$, a joint formed by them is a point lying on d given lines whose directions are linearly independent. The Joints Theorem gives a sharp upper bound of the total number of joints, up to a multiplicative constant. We present a new proof of the Joints Theorem without taking derivatives. Then we generalize our proof to prove the Multijoints Conjecture and Carbery’s generalization, which are unbalanced and weighted versions of the Joints Theorem. All results are in any dimension over an arbitrary field.