620 research outputs found
Distances on the tropical line determined by two points
Let . Write if is a multiple of
. Two different points and in uniquely
determine a tropical line , passing through them, and stable under
small perturbations. This line is a balanced unrooted semi--labeled tree on
leaves. It is also a metric graph.
If some representatives and of and are the first and second
columns of some real normal idempotent order matrix , we prove that the
tree is described by a matrix , easily obtained from . We also
prove that is caterpillar. We prove that every vertex in
belongs to the tropical linear segment joining and . A vertex, denoted
, closest (w.r.t tropical distance) to exists in . Same for
. The distances between pairs of adjacent vertices in and the
distances \dd(p,pq), \dd(qp,q) and \dd(p,q) are certain entries of the
matrix . In addition, if and are generic, then the tree
is trivalent. The entries of are differences (i.e., sum of principal
diagonal minus sum of secondary diagonal) of order 2 minors of the first two
columns of .Comment: 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. Jim\'enez; see Arxiv 1205.416
Tropical conics for the layman
We present a simple and elementary procedure to sketch the tropical conic
given by a degree--two homogeneous tropical polynomial. These conics are trees
of a very particular kind. Given such a tree, we explain how to compute a
defining polynomial.
Finally, we characterize those degree--two tropical polynomials which are
reducible and factorize them. We show that there exist irreducible degree--two
tropical polynomials giving rise to pairs of tropical lines.Comment: 19 pages, 4 figures. Major rewriting of formerly entitled paper
"Metric invariants of tropical conics and factorization of degree--two
homogeneous tropical polynomials in three variables". To appear in Idempotent
and tropical mathematics and problems of mathematical physics (vol. II), G.
Litvinov, V. Maslov, S. Sergeev (eds.), Proceedings Workshop, Moscow, 200
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