# The regularity lemma is false over small fields

## TL;DR

This paper demonstrates that the regularity lemma, which constrains tensor blow-up ranks over large fields, does not hold when the field size is too small, revealing limitations in small field contexts.

## Contribution

The paper proves that the regularity lemma fails over small fields, extending understanding of tensor rank behavior in different field sizes.

## Key findings

- Regularity lemma holds over large fields
- Fails over small fields
- Provides counterexamples for small fields

## Abstract

The regularity lemma is a stringent condition of the possible ranks of tensor blow-ups of linear subspaces of matrices. It was proved by Ivanyos, Qiao and Subrahmanyam when the underlying field is sufficiently large. We show that if the field size is too small, the regularity lemma is false.