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.

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.