Sign patterns of rational matrices with large rank

2013/12/18 by Shitov, Yaroslav
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1312.5369

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.

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