The 4x4 minors of a 5xn matrix are a tropical basis

TL;DR

This paper proves that for 5 by n matrices, the tropical rank and Kapranov rank are equal, and that the 4x4 minors form a tropical basis, resolving a question in tropical geometry.

Contribution

It establishes the equality of tropical and Kapranov ranks for 5 by n matrices and confirms the 4x4 minors form a tropical basis, answering a key open question.

Findings

Tropical rank equals Kapranov rank for 5 by n matrices.

4x4 minors form a tropical basis for these matrices.

Characterization of 5x5 matrices of tropical rank at most 3.

Abstract

We compute the space of 5x5 matrices of tropical rank at most 3 and show that it coincides with the space of 5x5 matrices of Kapranov rank at most 3, that is, the space of five labeled coplanar points in TP4. We then prove that the Kapranov rank of every 5xn matrix equals its tropical rank; equivalently, that the 4x4 minors of a 5xn matrix of variables form a tropical basis. This answers a question asked by Develin, Santos, and Sturmfels.