# Distances on the tropical line determined by two points

María Jesús de la Puente

# Abstract:

Let $p'$ and $q'$ be points in $\R^n$. Write $p'\sim q'$ if $p'-q'$ is a multiple of $(1,\ldots,1)$. Two different points $p$ and $q$ in $\R^n/\sim$ 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 $\dd(p,pq)$, $\dd(qp,q)$ and $\dd(p,q)$ are certain entries of the matrix $|F|$. 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 $A$.

# Keywords:

normal matrix, idempotent matrix, tropical distance, integer length, tropical line, caterpillar tree, metric graph

15A80, 14T05

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