由两点确定的热带线上的距离
Distances on the tropical line determined by two points
摘要：
令$p',q'\in R^n$。如果$p'-q'$是$(1,\ldots,1)$的倍数，则写入$p'\sim q'$。$R^n/\sim$中的两个不同点$p$和$q$唯一地确定了一条穿过它们的热带线$L(p,q)$，并且在小扰动下是稳定的。这条线是一个平衡的无根半标记树$n$叶。它也是一个度量图。如果$p$和$q$的某些代表$p'$和$q'$是某个实正规幂等阶$n$矩阵$A$的第一列和第二列，我们证明了树$L(p,q)$是由一个矩阵$F$描述的，这个矩阵很容易从$A$得到.我们还证明了$L(p,q)$是毛毛虫。我们证明了$L(p,q)$中的每个顶点都属于连接$p$和$q$的热带线段。$L(p,q)$中存在一个最接近（相对于热带距离）$p$的顶点，表示为$pq$。$q$也是如此。$L(p,q)$中相邻顶点对之间的距离以及距离$\dd(p,pq)$、$\dd(qp,q)$和$\dd(p,q)$是矩阵$|F|$的某些项。此外，如果$p$和$q$是通用的，则树$L(p,q)$是三价的。$F$的条目是差异（即，主对角线之和减去次对角线之和）。
英文摘要：
Let $p',q'\in R^n$. 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 $A$.
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