Rank-related dimension bounds for subspaces of symmetric bilinear forms

TL;DR

This paper explores bounds on the dimension of subspaces of symmetric bilinear forms based on rank conditions, revealing new bounds and structural insights, especially over fields of characteristic 2.

Contribution

It establishes new bounds on the dimension of subspaces of symmetric bilinear forms under rank restrictions, including characteristic-specific results and structural decompositions.

Findings

Bound dim M by r(r+1)/2 when all non-zero elements have odd rank r

Dim M ≤ r when the field is finite or of characteristic 2

Structural decompositions related to spreads, pseudo-arcs, and pseudo-ovals for certain rank and dimension conditions

Abstract

Let V be a vector space of dimension n over a field K and let Symm(V) denote the space of symmetric bilinear forms defined on V x V. Let M be a subspace of Symm(V). We investigate a variety of hypotheses concerning the rank of elements in M that lead to reasonable bounds for dim M. For example, if every non-zero element of M has odd rank, and r is the maximum rank of the elements of M, then dim M is at most r(r+1)/2 (thus dim M is bounded independently of n). This should be contrasted with the simple observation that Symm(V) contains a subspace of dimension n-1 in which each non-zero element has rank 2. The bound r(r+1)/2 is almost certainly too large, and a bound r seems plausible, this being true when K is finite. We also show that dim M is at most r$ when K is any field of characteristic 2. Finally, suppose that n=2r, where r is an odd integer, and the rank of each non-zero element of M is either r or n. We show that if K has characteristic 2, then dim M is at most 3r. Furthermore, if dim M=3r, we obtain interesting subspace decompositions of M and V related to spreads, pseudo-arcs and pseudo-ovals. Examples of such subspaces M exist if K has an extension field of degree r.