Point-Line Geometry in the Tropical Plane

TL;DR

This paper extends classical incidence geometry results to the tropical plane, introducing stable tropical lines and utilizing duality principles to establish bounds on line configurations.

Contribution

It introduces stable tropical lines and applies tropical duality to derive bounds on line configurations in the tropical plane, advancing tropical incidence geometry.

Findings

Bound on the number of stable tropical lines determined by n points

Classification of stable intersections via Newton subdivisions

Insights into properties of linear Newton subdivisions

Abstract

We study the classical result by Bruijn and Erd\H os regarding the bound on the number of lines determined by a $n$-point configuration in the plane, and in the light of the recently proven Tropical Sylvester-Gallai theorem, come up with a tropical version of the above-mentioned result. In this work, we introduce stable tropical lines, which help in answering questions pertaining to incidence geometry in the tropical plane. Projective duality in the tropical plane helps in translating the question for stable lines to stable intersections that have been previously studied in depth. Invoking duality between Newton subdivisions and line arrangements, we are able to classify stable intersections with shapes of cells in subdivisions, and this ultimately helps us in coming up with a bound. In this process, we also encounter various unique properties of linear Newton subdivisions which are dual to tropical line arrangements.