Tropical linear maps on the plane

✍ Scribed by M.J. de la Puente

• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 922 KB
• Volume: 435
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2010.07.031

✦ Synopsis

In this paper we fully describe all tropical linear maps in the tropical projective plane TP 2 , that is, maps from the tropical plane to itself given by tropical multiplication by a real 3 × 3 matrix A. The map f A is continuous and piecewise-linear in the classical sense. In some particular cases, the map f A is a parallel projection onto the set spanned by the columns of A. In the general case, after a change of coordinates, the map 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 f A , 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 lower canonical normalization) (Theorem 1) and then always work with it, due both to its algebraic simplicity and its geometrical meaning.