Incidence geometry and universality in the tropical plane

TL;DR

This paper explores the incidence geometry of tropical lines, establishing tropical versions of classical theorems and revealing that realizability of tropical line arrangements is as complex as solving arbitrary integer linear programs.

Contribution

It introduces tropical analogs of key classical theorems and demonstrates the computational complexity of realizability problems in tropical geometry.

Findings

Tropical Sylvester-Gallai and Motzkin-Rabin theorems proved

Realizability of tropical line arrangements is NP-hard

Connections to universality theorems in algebraic geometry

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 Mn\"ev'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.