# Rank-related dimension bounds for subspaces of bilinear forms over finite fields

## TL;DR

This paper explores bounds on the dimension of subspaces of bilinear forms over finite fields based on the ranks of their elements, focusing on cases with limited rank variability and geometric properties of radicals.

## Contribution

It establishes rank-related dimension bounds for subspaces of bilinear forms and characterizes forms with maximal dimension, including their geometric properties.

## Key findings

- Derived bounds for subspace dimensions based on rank constraints
- Enumerated forms of specific ranks within maximal subspaces
- Described geometric properties of radicals of degenerate forms

## Abstract

Let q be a power of a prime and let V be a vector space of finite dimension n over the field of order q. Let Bil(V) denote the set of all bilinear forms defined on V x V, let Symm(V) denote the subspace of Bil(V) consisting of symmetric bilinear forms, and Alt(V) denote the subspace of alternating bilinear forms. Let M denote a subspace of any of the spaces Bil(V), Symm(V), or Alt(V). In this paper we investigate hypotheses on the rank of the non-zero elements of M which lead to reasonable bounds for dim M. Typically, we look at the case where exactly two or three non-zero ranks occur, one of which is usually n. In the case that M achieves the maximal dimension predicted by the dimension bound, we try to enumerate the number of forms of a given rank in M and describe geometric properties of the radicals of the degenerate elements of M.