Rank-related dimension bounds for subspaces of symmetric bilinear forms

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.