Large spaces of bounded rank matrices revisited

TL;DR

This paper classifies large affine subspaces of matrices with bounded rank over all fields, extending known results and providing a comprehensive understanding of their structure.

Contribution

It introduces a new method to classify large rank-bounded affine subspaces over all fields, doubling the known dimension range for such structures.

Findings

Classified large affine subspaces of matrices with bounded rank over all fields.

Extended the known dimension range for rank-$ar{r}$ linear subspaces.

Provided a comprehensive structural understanding of these subspaces.

Abstract

Let $n,p,r$ be positive integers with $n \geq p\geq r$. A rank-$\overline{r}$ subset of $n$ by $p$ matrices (with entries in a field) is a subset in which every matrix has rank less than or equal to $r$. A classical theorem of Flanders states that the dimension of a rank-$\overline{r}$ linear subspace must be less than or equal to $nr$, and it characterizes the spaces with the critical dimension $nr$. Linear subspaces with dimension close to the critical one were later studied by Atkinson, Lloyd and Beasley over fields with large cardinality; their results were recently extended to all fields. Using a new method, we obtain a classification of rank-$\overline{r}$ affine subspaces with large dimension, over all fields. This classification is then used to double the range of (large) dimensions for which the structure of rank $\overline{r}$-linear subspaces is known for all fields.