Distances on the tropical line determined by two points
type: Item
instance of: scholarly article
zbMATH Open document ID: 1321.14050
author: María Jesús de la Puente
publication date: 2014-10-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1310.0174
full work available at URL: http://www.kybernetika.cz/content/2014/3/408
Mathematics Subject Classification ID: 14T05
Mathematics Subject Classification ID: 15A80
zbMATH DE Number: 6357558
zbMATH Keywords: tropical distance
zbMATH Keywords: integer length
zbMATH Keywords: tropical line
zbMATH Keywords: normal matrix
zbMATH Keywords: idempotent matrix
zbMATH Keywords: caterpillar tree
zbMATH Keywords: metric graph
MaRDI profile type: Publication
arXiv ID: 1310.0174
arXiv classification: math.MG
arXiv classification: math.CO
arXiv classification: math.RA
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title: Distances on the tropical line determined by two points
published in: Kybernetika
review text: Let \(n\geq 3\). Consider the tropical projective torus \(Q^{n-1}=\mathbb R^n/\mathbb R(1,\dots, 1)\), endowed with the tropical distance \(d(p,q)=\max |p_i-q_i-p_j+q_j|\). Let \(L\) be the unique continuous map that assigns to each pair \((p,q)\) of distinct points in \(Q^{n-1}\) a tropical line \(L(p,q)\) passing through them. This line is a metric tree on \(n\) leaves. It contains the tropical segment NEWLINE\[NEWLINE\mathrm{tconv}(p,q)=\{\lambda \odot p\oplus \mu\odot q\quad|\quad\lambda, \mu\in\mathbb R\}.NEWLINE\]NEWLINENEWLINENEWLINEThe paper under review describes the line \(L(p,q)\) if \(p\) and \(q\) are (represented by) the first two columns of some real, normal, tropically idempotent \(n\)-square matrix, where `normal' means that all diagonal entries are zero and all entries are non-positive. (Tropical addition is defined to be maximum, not minimum.) It is proved that every vertex of \(L(p,q)\), excepting the leaves, belongs to the tropical linear segment joining \(p\) and \(q\), which may fail under less restrictive conditions. Moreover, \(L(p,q)\) is a caterpillar tree, i.e., it contains a path passing through all vertices except the leaves. The distances between adjacent vertices, and also those between \(p\) (resp.\ \(q\)) and the nearest vertex, are of the form \( |p_i-q_i-p_j+q_j|\). In addition, if \(p\) and \(q\) are generic, then the tree \(L(p,q)\) is trivalent.
reviewed by: Péter E. Frenkel
description: scientific article; zbMATH DE number 6357558