# Distances on the tropical line determined by two points

### Impacto

Puente Muñoz, María Jesús de la (2014) Distances on the tropical line determined by two points. Kybernetika, 50 (3). pp. 408-435. ISSN 0023-5954

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http://arxiv.org/abs/1310.0174Organisation

## Abstract

Let p' and q' be points in R-n. Write p' similar to q' if p' - q' is a multiple of (1,...,1). Two different points p and q in R-n/ similar to uniquely determine a tropical line L(p, q) passing through them and stable under small perturbations. This line is a balanced unrooted semi-labeled 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 tropical distance; integer length; tropical line; normal matrix; idempotent matrix; caterpillar tree; metric graph Sciences > Mathematics > Algebraic geometry 27290 06 Nov 2014 08:41 20 Jan 2016 15:07