Partial matrices of constant rank

Abstract

A partial matrix over a field F is a matrix whose entries are either elements of F or independent indeterminates. A completion of a partial matrix is obtained by specifying values from F 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 F possesses an r×r partial submatrix of constant rank r if and only if |F|⩾r. If |F|<r, we show that there exist counterexamples of size m×n to this assertion provided that max(m,n)⩾r+|F|−1.