On matrices with different tropical and Kapranov ranks

TL;DR

This paper investigates the relationship between tropical and Kapranov ranks of matrices, proving bounds for certain sizes and providing counterexamples over specific fields, thus advancing understanding in tropical linear algebra.

Contribution

It generalizes existing techniques to show that 5xn matrices with tropical rank ≤ 3 have Kapranov rank ≤ 3 over fields with at least 4 elements, and constructs counterexamples over F_2 and F_3.

Findings

5xn matrices with tropical rank ≤ 3 have Kapranov rank ≤ 3 over suitable fields

Counterexamples exist over F_2 and F_3 with tropical rank 3 and Kapranov rank 4

The technique from [13] is effectively generalized for these results

Abstract

In this note, we generalize the technique developed in [13] and prove that every 5xn matrix of tropical rank at most 3 has Kapranov rank at most 3, for the ground field that contains at least 4 elements. For the ground field either F_2 or F_3, we construct an example of a 5x5 matrix with tropical rank 3 and Kapranov rank 4.