A separation between tropical matrix ranks

2017/12/08 by Shitov, Yaroslav
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1712.03071

Abstract

We continue to study the rank functions of tropical matrices. In this paper, we explain how to reduce the computation of ranks for matrices over the `supertropical semifield' to the standard tropical case. Using a counting approach, we prove the existence of a 01-matrix with many ones and without large all-one submatrices, and we put our results together and construct an n× n matrix with tropical rank o(n^0.5+ε) and Kapranov rank n-o(n).

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• On the rank of a tropical matrix
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• The ultimate rank of tropical matrices