Title: Rank-related dimension bounds for subspaces of bilinear forms over finite fields,
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.