A note on tropical triangles in the plane

Abstract

We define transversal tropical triangles (affine and projective) and characterize them via six inequalities to be satisfied by the coordinates of the vertices. We prove that the vertices of a transversal tropical triangle are tropically independent and they tropically span a classical hexagon whose sides have slopes $\infty ,0,1$. Using this classical hexagon, we determine a parameter space for transversal tropical triangles. The coordinates of the vertices of a transversal tropical triangle determine a tropically regular matrix. Triangulations of the tropical plane are obtained.

15 pages, 2 figures. The paper has been fully revised and enlarged. Tropical triangles called transversal here, were called proper and stable in a previous version. To appear in Acta Math. Sinica (English series)

References in corpus (5)

• The Maslov dequantization, idempotent and tropical mathematics: A brief introduction
• Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry
• The group law on a tropical elliptic curve
• Duality of Tropical Curves
• Tropical Plane Geometric Constructions: a Transfer Technique in Tropical Geometry