On tropical and Kapranov ranks of tropical matrices

TL;DR

This paper proves that for matrices with at least 3 rows and 5 columns, having tropical rank 3 guarantees Kapranov rank 3, establishing a rank equivalence in this case.

Contribution

It establishes a new rank equivalence result for tropical and Kapranov ranks in matrices with specific dimensions.

Findings

Tropical rank 3 implies Kapranov rank 3 for matrices with at least 3 rows and 5 columns.

The result holds for any number of rows g ≥ 3.

Clarifies the relationship between tropical and Kapranov ranks in this matrix class.

Abstract

We prove that, for any g greater or equal than 3, a matrix g x 5 with tropical rank 3 has Kapranov rank 3.