A proof of the Multijoints Conjecture and Carbery's generalization

TL;DR

The paper provides a derivative-free proof of the Joints Theorem and extends it to prove the Multijoints Conjecture and Carbery's generalization across all dimensions and fields.

Contribution

It introduces a novel proof technique that avoids derivatives and generalizes key geometric combinatorics conjectures.

Findings

Proof of the Joints Theorem without derivatives

Extension to the Multijoints Conjecture

Generalization to arbitrary fields and dimensions

Abstract

We present a new proof of the Joints Theorem without taking derivatives. Then we generalize the proof to prove the Multijoints Conjecture and Carbery's generalization. All results are in any dimension over an arbitrary field.