On the maximum rank of completions of entry pattern matrices

In an entry pattern matrix A, all entries are indeterminates but the same indeterminate can appear in numerous positions. For a field F, an F-completion of A results from assigning a value from F to each indeterminate entry. We define the generic F-rank of an entry pattern matrix to be its rank when considered over the function field generated over F by its indeterminate entries. We investigate the situation where the generic F-rank of A is not attained by any F-completion of A, which can occur only if the generic F-rank exceeds the field order.