The \(4\times 4\) minors of a \(5\times n\) matrix are a tropical basis
type: Item
instance of: scholarly article
title: The \(4\times 4\) minors of a \(5\times n\) matrix are a tropical basis
zbMATH Open document ID: 1231.14053
author: Melody Chan
author: Elena Rubei
author: Anders Nedergaard Jensen
published in: Linear Algebra and its Applications
publication date: 2011-07-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/0912.5264
review text: One of the main difficulties in tropical geometry is caused by the fact that the intersection of the tropical hypersurfaces defined by a set of polynomials can be strictly larger than the tropical variety of the ideal that they generate. A particular case of this phenomenon is that the tropical rank of a matrix can be strictly smaller than its Kapranov rank. Indeed, it is known that there exists a 7-square matrix with tropical rank 3 and Kapranov rank 4 [\textit{M. Develin, F. Santos, B. Sturmfels}, in: Combinatorial and computational geometry. Cambridge: Cambridge University Press. Mathematical Sciences Research Institute Publications 52, 213--242 (2005; Zbl 1095.15001)]. However, in that paper it is proved that for \(4\times n\) matrices those two notions of rank coincide, and it is asked whether this holds for 5-square matrices as well. In the paper under review, the authors compute the space of 5-square matrices of tropical rank at most 3 and show that it coincides with the space of 5-square matrices of Kapranov rank at most 3, that is, the space of five labeled coplanar points in the tropical torus. They then prove that the Kapranov rank of every \(5\times n\) matrix equals its tropical rank; equivalently, that the \(4\times 4\) minors of a \(5\times n\) matrix of variables form a tropical basis. A less elegant proof of this result had previously been given in a preprint by the third author [\textit{E. Rubei}, On tropical and Kapranov ranks of tropical matrices, \url{arXiv:0712.3007}].
reviewed by: Péter E. Frenkel
Mathematics Subject Classification ID: 14T05
zbMATH DE Number: 5919612
zbMATH Keywords: Kapranov rank
zbMATH Keywords: tropical rank
zbMATH Keywords: tropical bases
describes a project that uses: cdd
describes a project that uses: Gfan
describes a project that uses: SoPlex
MaRDI profile type: Publication
OpenAlex ID: W1482907961
arXiv ID: 0912.5264
cites work: Q5290261
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DOI: 10.1016/J.LAA.2010.09.032
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description: scientific article