[Paper Review] The 4x4 minors of a 5xn matrix are a tropical basis

Melody Chan, Anders Jensen+1 more|ArXiv.org|Dec 29, 2009

Polynomial and algebraic computation12 references4 citations

TL;DR

This paper proves that the 4×4 minors of a 5×n matrix form a tropical basis, meaning the tropical rank and Kapranov rank of any 5×n matrix are equal. Using polyhedral geometry and computational algebraic techniques via Gfan, the authors show that the space of 5×5 matrices of tropical rank ≤3 coincides with those of Kapranov rank ≤3, resolving a question by Develin, Santos, and Sturmfels. The result extends to all 5×n matrices, establishing a fundamental equivalence in tropical matrix theory.

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.

Motivation & Objective

• To determine whether the 4×4 minors of a 5×n matrix form a tropical basis, which would imply that tropical rank and Kapranov rank coincide for such matrices.
• To resolve a question posed by Develin, Santos, and Sturmfels regarding the tropical basis property of 4×4 minors in the 5×n case.
• To compute and compare the polyhedral fans of matrices of tropical rank ≤3 and Kapranov rank ≤3 in the 5×5 case, showing their supports are identical.
• To generalize the result from 5×5 to 5×n matrices using structural and combinatorial arguments, including column development techniques.
• To establish that for d or n ≤ 5, tropical rank and Kapranov rank are always equal, providing a broad class of matrices where these notions coincide.

Proposed method

• Computing the space of 5×5 matrices of tropical rank ≤3 as a polyhedral fan by intersecting tropical hypersurfaces defined by 4×4 minors.
• Using the software Gfan to compute the Gröbner fan of the ideal generated by 4×4 minors of a 5×5 matrix, which corresponds to the Kapranov rank ≤3 locus.
• Comparing the two fans (tropical rank ≤3 and Kapranov rank ≤3) by verifying they have identical supports, confirming they define the same set of matrices.
• Applying a column-wise 'development by a column' technique to analyze matrix rank equivalence without relying solely on computation, especially for the 5×n case.
• Using witness points and type analysis for hyperplanes in tropical projective space to verify non-stable intersections and derive contradictions in cases where rank discrepancy might occur.
• Employing rescaling and translation in tropical projective space to normalize configurations and simplify inequalities governing rank conditions.

Experimental results

Research questions

• RQ1 Do the 4×4 minors of a 5×5 matrix form a tropical basis?
• RQ2 Is the tropical rank of a 5×n matrix always equal to its Kapranov rank?
• RQ3 Does the space of 5×5 matrices of tropical rank ≤3 coincide with the space of those of Kapranov rank ≤3?
• RQ4 Can the tropical basis property of 4×4 minors be extended from 5×5 to 5×n matrices?
• RQ5 What is the geometric relationship between the tropical rank and Kapranov rank in low-dimensional matrix settings?

Key findings

• The 4×4 minors of a 5×n matrix form a tropical basis, meaning the tropical variety defined by the ideal of 4×4 minors equals the intersection of the tropical hypersurfaces defined by each minor.
• The space of 5×5 matrices of tropical rank ≤3 is identical to the space of 5×5 matrices of Kapranov rank ≤3, as confirmed by fan comparison using Gfan.
• For any 5×n matrix, the tropical rank equals the Kapranov rank, which implies that the 4×4 minors form a tropical basis for all n ≥ 4.
• The proof of the 5×n case relies on a structural argument using 'development by a column' and type analysis of witnesses to non-stable intersections of tropical hyperplanes.
• The authors resolve a gap in an earlier version of the proof and extend the argument to the general 5×n case, confirming the result for all n ≥ 4.
• The result provides a complete characterization of when tropical rank and Kapranov rank coincide, showing they are equal for all d×n matrices with d ≤ 5 or n ≤ 5.