# A dimension bound for subspaces of symmetric bilinear forms in terms of the number of different ranks

## TL;DR

This paper establishes an upper bound on the dimension of subspaces of symmetric bilinear forms based on the number of distinct ranks present, under certain field size conditions.

## Contribution

It introduces a new inequality relating the dimension of symmetric bilinear form subspaces to the number of different ranks, extending understanding of their structure.

## Key findings

- Derived an upper bound for the dimension of subspaces in terms of rank diversity.
- Provided conditions on the field size for the inequality to hold.
- Enhanced theoretical understanding of symmetric bilinear form subspaces.

## Abstract

Let K be a field of characteristic different from 2 and let V be a vector space of dimension n over K. Let M be a non-zero subspace of symmetric bilinear forms defined on V x V and let r=rank(M) denote the set of different positive integers that occur as the ranks of the non-zero elements of M. The main result of this paper is the inequality that dim M is at most rn-r(r-1)/2 provided that |K| is at least n.

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Source: https://tomesphere.com/paper/1704.07634