Distances on the tropical line determined by two points

الأصل · EN

Let p',q'∈ Rⁿ. Write p' q' if p'-q' is a multiple of (1,,1). Two different points p and q in Rⁿ/ 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 (p,pq), (qp,q) and (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.