Partial matrices of constant rank

A partial matrix over a field FF is a matrix whose entries are either elements of FF or independent indeterminates. A completion of a partial matrix is obtained by specifying values from FF for the indeterminates. A partial matrix of constant rank is one whose completions all have the same rank. We show that every partial matrix of constant rank r   over FF possesses an r×rr×r partial submatrix of constant rank r   if and only if |F|⩾r|F|⩾r. If |F|