A separation between tropical matrix ranks

TL;DR

This paper explores the differences between various tropical matrix ranks, providing methods to compute and compare them, and constructs matrices demonstrating significant rank disparities.

Contribution

It introduces a reduction technique for tropical ranks over the supertropical semifield and constructs matrices with large Kapranov rank but small tropical rank.

Findings

Existence of a 01-matrix with many ones but no large all-one submatrix

Construction of an n×n matrix with tropical rank o(n^{0.5+ε}) and Kapranov rank n-o(n)

Reduction method for computing ranks over the supertropical semifield

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 $n\times n$ matrix with tropical rank $o(n^{0.5+\varepsilon})$ and Kapranov rank $n-o(n)$.