Incidence geometry and universality in the tropical plane

Abstract

We examine the incidence geometry of lines in the tropical plane. We prove tropical analogs of the Sylvester–Gallai and Motzkin–Rabin theorems in classical incidence geometry. This study leads naturally to a discussion of the realizability of incidence data of tropical lines. Drawing inspiration from the von Staudt constructions and Mnëv's universality theorem, we prove that determining whether a given tropical linear incidence datum is realizable by a tropical line arrangement requires solving an arbitrary linear programming problem over the integers.