On the Kapranov ranks of tropical matrices
type: Item
instance of: scholarly article
title: On the Kapranov ranks of tropical matrices
zbMATH Open document ID: 1243.15013
published in: Linear Algebra and its Applications
publication date: 2012-05-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1207.4697
review text: The author investigates the Kapranov rank functions of tropical matrices for different ground fields. For any infinite ground field he shows that the rank-product inequality holds for the Kapranov rank, and he proves that the Kapranov rank respects Green's preorders on the semigroup of tropical \(n\)-by-\(n\) matrices. The rank-product inequality is shown to fail for the Kapranov rank over any finite ground field. He provides an example of a 7-by-7 0-1-matrix whose Kapranov rank is independent of the ground field, equals 6, and exceeds tropical rank.
Mathematics Subject Classification ID: 15A80
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 16Y60
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 15A45
zbMATH DE Number: 6034453
zbMATH Keywords: tropical semiring
zbMATH Keywords: tropical rank
zbMATH Keywords: Kapranov rank
zbMATH Keywords: tropical matrices
zbMATH Keywords: infinite ground field
zbMATH Keywords: rank-product inequality
zbMATH Keywords: semigroup
author: Ya. N. Shitov
reviewed by: Andreas Arvanitoyeorgos
MaRDI profile type: Publication
OpenAlex ID: W2065371460
arXiv ID: 1207.4697
cites work: Linear independence over tropical semirings and beyond
cites work: RANK INEQUALITIES OVER SEMIRINGS
cites work: A Catalogue of Combinatorial Geometries
cites work: The \(4\times 4\) minors of a \(5\times n\) matrix are a tropical basis
cites work: Q5290261
cites work: Kapranov rank vs. tropical rank
cites work: Q5315152
cites work: Example of the matrix of size \((6\times 6)\) with different tropical rank and Kapranov rank
cites work: Inequalities for Gondran-Minoux rank and idempotent semirings
DOI: 10.1016/J.LAA.2011.11.004
description: scientific article; zbMATH DE number 6034453