Distances on the tropical line determined by two points

Abstract

Let $p^{\prime },q^{\prime }\in R^n$. Write $p^{\prime }\sim q^{\prime }$ if $p^{\prime }-q^{\prime }$ is a multiple of $(1,\dots ,1)$. 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 $p^{\prime }$ and $q^{\prime }$ 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 $A$.

arXiv commentNew corrected version. 31 pages and 9 figures. The main result is theorem 13. This is a generalization of theorem 7 to arbitrary n. Theorem 7 was obtained with A. Jiménez; see Arxiv 1205.4162