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Puente Muñoz, María Jesús de la (2014) Distances on the tropical line determined by two points. Kybernetika, 50 (3). pp. 408435. ISSN 00235954

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Official URL: http://www.kybernetika.cz/content/2014/3/408/paper.pdf
URL  URL Type 

http://arxiv.org/abs/1310.0174  Organisation 
Abstract
Let p' and q' be points in Rn. Write p' similar to q' if p'  q' is a multiple of (1,...,1). Two different points p and q in Rn/ similar to uniquely determine a tropical line L(p, q) passing through them and stable under small perturbations. This line is a balanced unrooted semilabeled tree on n leaves. It is also a metric graph.
If some representatives p' and q' of p and q are the first and second columns of some real normal idempotent order n matrix A, we prove that the tree L(p, q) is described by a matrix F, easily obtained from A. We also prove that L(p, q) is caterpillar. We prove that every vertex in L(p, q) belongs to the tropical linear segment joining p and q. A vertex, denoted pq, closest (w.r.t tropical distance) to p exists in L(p, q). Same for q. The distances between pairs of adjacent vertices in L(p, q) and the distances d(p, pq), d(qp, q) and d(p, q) are certain entries of the matrix vertical bar F vertical bar. In addition, if p and q are generic, then the tree L(p, q) is trivalent. The entries of F are differences (i.e., sum of principal diagonal minus sum of secondary diagonal) of order 2 minors of the first two columns of A.
Item Type:  Article 

Uncontrolled Keywords:  tropical distance; integer length; tropical line; normal matrix; idempotent matrix; caterpillar tree; metric graph 
Subjects:  Sciences > Mathematics > Algebraic geometry 
ID Code:  27290 
Deposited On:  06 Nov 2014 08:41 
Last Modified:  20 Jan 2016 15:07 
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