@MISC{Puente09tropicallinear, author = {M. J. De La Puente}, title = {Tropical linear mappings on the plane}, year = {2009} }

Share

OpenURL

Abstract

In this paper we fully describe all tropical linear mappings in the tropical projective plane TP 2, that is, maps from the tropical plane to itself given by tropical multiplication by a 3 × 3 matrix A with entries in T. First we will allow only real entries in the matrix A and, only at the end of the paper, we will allow some of the entries of A equal −∞. The mapping fA is continuous and piecewise–linear in the classical sense. In some particular cases, the mapping fA is a parallel projection onto the set spanned by the columns of A. In the general case, after a change of coordinates, the mapping collapses at most three regions of the plane onto certain segments, called antennas, and is a parallel projection elsewhere (theorem 3). In order to study fA, we may assume that A is normal, i.e., I ≤ A ≤ 0, up to changes of coordinates. A given matrix A admits infinitely many normalizations. Our approach is to define and compute a unique normalization for A (which we call canonical normalization) (theorem 1) and then always work with it, due both to its algebraic simplicity and its geometrical meaning.