2013/10/01 by M. J. de la Puente, de la Puente, M. J.
Mathematics · #14T05 #15A80 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Rings and Algebras (math.RA) #math.CO #math.MG #math.RA #msc:14T05 #msc:15A80
paper · pdf · doi:10.48550/arxiv.1310.0174
New 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
arxiv created 2014/04/10 · arxiv updated 2014/04/11
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 \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.