Connections between rank and dimension for subspaces of bilinear forms

TL;DR

This paper investigates upper bounds on the dimension of subspaces of bilinear forms based on the number of distinct non-zero ranks, providing new bounds under various conditions and conjectures for constant rank spaces.

Contribution

It establishes new upper bounds for the dimension of subspaces of bilinear forms related to their rank properties, including proofs for conjectures in specific cases.

Findings

For m ≤ ⌈n/2⌉ and |K| ≥ m+1, dim M ≤ r n.

If M has constant rank m, then dim M ≤ max(n, 2m-1) under certain conditions.

Provides detailed results for subspaces over finite fields, especially symmetric and alternating forms.

Abstract

Let $K$ be a field and let $V$ be a vector space of dimension $n$ over $K$. Let $M$ be a subspace of bilinear forms defined on $V\times V$. Let $r$ be the number of different non-zero ranks that occur among the elements of $M$. Our aim is to obtain an upper bound for $\dim M$ in terms of $r$ and $n$ under various hypotheses. As a sample of what we prove, we mention the following. Suppose that $m$ is the largest integer that occurs as the rank of an element of $M$. Then if $m\leq \lceil n/2\rceil$ and $|K|\geq m+1$, we have $\dim M\leq rn$. The case $r=1$ corresponds to a constant rank space and it is conjectured that $\dim M\leq n$ when $M$ is a constant rank $m$ space and $|K|\geq m+1$. We prove that the dimension bound for a constant rank $m$ space $M$ holds provided $|K|\geq m+1$ and either $K$ is finite or $K$ has characteristic different from 2 and $M$ consists of symmetric forms. In general, we show that if $M$ is a constant rank $m$ subspace and $|K|\geq m+1$, then $\dim M\leq \max\,(n,2m-1)$. We also provide more detailed results about constant rank subspaces over finite fields, especially subspaces of alternating or symmetric bilinear forms.