Counting joints in vector spaces over arbitrary fields

TL;DR

This paper extends the joint counting theorem to vector spaces over any field, not just real numbers, and discusses multiplicity estimates for generic line families.

Contribution

It generalizes the folklore joint counting theorem to arbitrary fields and provides distributional estimates for joint multiplicities in generic cases.

Findings

Joint counting theorem holds over arbitrary fields.

Distributional estimates for joint multiplicities.

Applicable to generic line configurations.

Abstract

We give a proof of the "folklore" theorem that the Kaplan--Sharir--Shustin/Quilodr\'an result on counting joints associated to a family of lines holds in vector spaces over arbitrary fields, not just the reals. We also discuss a distributional estimate on the multiplicities of the joints in the case that the family of lines is sufficiently generic.