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

2009/12/29 by Melody Chan, Chan, Melody, Anders Jensen +3 · 1 citation
Computer Science · Engineering · #14T05 #Combinatorics (math.CO) #FOS: Mathematics #Matrix Theory and Algorithms #Polynomial and algebraic computation #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.0912.5264

openalex publication_date 2009/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• The Tropical Grassmannian

Cited by

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