Connections between rank and dimension for subspaces of bilinear forms

Original · EN

Let K be a field and let V be a vector space of dimension n over K. Let M be a subspace of bilinear forms defined on V× V. Let r be the number of different non-zero ranks that occur among the elements of M. Our aim is to obtain an upper bound for M in terms of r and n under various hypotheses. As a sample of what we prove, we mention the following. Suppose that m is the largest integer that occurs as the rank of an element of M. Then if m≤ n/2 and |K|≥ m+1, we have M≤ rn. The case r=1 corresponds to a constant rank space and it is conjectured that M≤ n when M is a constant rank m space and |K|≥ m+1. We prove that the dimension bound for a constant rank m space M holds provided |K|≥ m+1 and either K is finite or K has characteristic different from 2 and M consists of symmetric forms. In general, we show that if M is a constant rank m subspace and |K|≥ m+1, then M≤ (n,2m-1). We also provide more detailed results about constant rank subspaces over finite fields, especially subspaces of alternating or symmetric bilinear forms.