Sign patterns of rational matrices with large rank

Abstract

Let A be a real matrix. The term rank of A is the smallest number t of lines (that is, rows or columns) needed to cover all the nonzero entries of A. We prove a conjecture of Li et al. stating that, if the rank of A exceeds t−3, there is a rational matrix with the same sign pattern and rank as those of A. We point out a connection of the problem discussed with the Kapranov rank function of tropical matrices, and we show that the statement fails to hold in general if the rank of A does not exceed t−3.