ar X iv : 0 90 7 . 28 11 v 1 [ m at h . A G ] 1 6 Ju l 2 00 9 Tropical linear mappings on the plane

In this paper we fully describe all tropical linear mappings in the tropical pro-jective 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. On R n , any n ∈ N, some aspects of tropical lineal maps have been studied in [5]. We work in TP 2 , adding a geometric view and doing everything explicitly. We give precise pictures. Inspiration for this paper comes from [3, 5, 7, 11, 24]. We have tried to make it self–contained. Our preparatory results present noticeable relationships between the algebraic properties of a given matrix A (normal idempotent matrix, permutation matrix, etc.) and classical geometric properties of the points spanned by the columns of A (classical convexity and others); see theorem 2 and corollary 1. As a by–product, we compute all the tropical square roots of normal matrices of a certain type; see corollary 3. This is, perhaps, a curious result in tropical algebra. Our final aim is, however, to give a precise description of the mapping fA : TP 2 → TP 2. This is particularly easy when two tropical triangles arising from A (denoted TA and T A) fit as much as possible. Then the action of fA is easily described on each cell of the cell decomposition C A ; see theorem 3. Normal matrices play a crucial role in this paper. …