When do the \(r\)-by-\(r\) minors of a matrix form a tropical basis?
type: Item
instance of: scholarly article
title: When do the \(r\)-by-\(r\) minors of a matrix form a tropical basis?
zbMATH Open document ID: 1300.14065
published in: Journal of Combinatorial Theory. Series A
publication date: 2013-12-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1109.2240
reviewed by: Péter E. Frenkel
Mathematics Subject Classification ID: 14T05
zbMATH DE Number: 6239411
zbMATH Keywords: tropical basis
zbMATH Keywords: tropical rank
zbMATH Keywords: Kapranov rank
author: Ya. N. Shitov
MaRDI profile type: Publication
OpenAlex ID: W2010122474
arXiv ID: 1109.2240
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DOI: 10.1016/J.JCTA.2013.03.004
review text: Let \(A\) be a real matrix of size \(6\times n\) whose tropical rank is at most 3, i.e., among the \(4!\) four-term sums defining the tropical permanent of any 4-square submatrix, the minimum is attained at least twice. In the paper under review, it is proved that then \(A\) has Kapranov rank at most 3, i.e., \(A\) can be lifted to a matrix \(\tilde A\) of Hahn series such that \(\tilde A\) has rank at most 3.NEWLINENEWLINELet \(n\geq d\) be positive integers throughout what follows. It has been shown in [\textit{M. Develin} et al., in: Combinatorial and computational geometry. Cambridge: Cambridge University Press. Mathematical Sciences Research Institute Publications 52, 213--242 (2005; Zbl 1095.15001)] that the Kapranov rank of any \(d\times n\) matrix is at least its tropical rank, and the two ranks coincide if the tropical rank is at most 2 or the Kapranov rank is \(d\). Later, [\textit{M. Chan} et al., Linear Algebra Appl. 435, No. 7, 1598--1611 (2011; Zbl 1231.14053)] proved that the two ranks coincide if \(d\leq 5\). \textit{Ya. Shitov} in a previous work [``Example of a 6-by-6 matrix with different tropical and Kapranov ranks'', Vestnik Moskov. Univ. Ser. I 5, 58--61 (2011) (in Russian); English version available via \url{arXiv:1012.5507}] gave an example of a 6-square matrix with tropical rank 4 and Kapranov rank 5. As remarked in the paper under review, this trivially yields a \(d\times n\) matrix of tropical rank 4 and Kapranov rank 5 whenever \(d\geq 6\). Already in [Zbl 1095.15001], an example of of a \(7\)-square matrix with tropical rank 3 and Kapranov rank 4 was given. In the paper under review, it is explained how to modify these examples to get a \(d\times n\) matrix of tropical rank less than \(r\) but Kapranov rank at least \(r\) whenever \(4\leq r<d\geq 7\).NEWLINENEWLINEIt is also explained that the existence of such an example for given \(r\), \(d\), \(n\) is equivalent to saying that the \(r\)-square minors of a \(d\times n\) matrix of algeb…
description: scientific article; zbMATH DE number 6239411