Tropical Plane Geometric Constructions: a Transfer Technique in Tropical Geometry

TL;DR

This paper introduces a transfer technique in tropical geometry that enables the translation of classical geometric theorems into the tropical setting by establishing conditions for algebraic counterparts of tropical constructions.

Contribution

It develops an algorithm to find conditions for algebraic counterparts of tropical geometric constructions and applies this to transfer classical theorems into tropical geometry.

Findings

Algorithm for algebraic counterparts of tropical constructions

Tropical versions of classical theorems like Pascal, Fano, Cayley-Bacharach

Conditions ensuring the existence of algebraic counterparts

Abstract

The notion of geometric construction is introduced. This notion allows to compare incidence configurations in the algebraic and tropical plane. We provide an algorithm such that, given a tropical instance of a geometric construction, it computes sufficient conditions to have an algebraic counterpart related by tropicalization. We also provide sufficient conditions in a geometric construction to ensure that the algebraic counterpart always exists. Geometric constructions are applied to transfer classical theorems to the tropical framework, we provide a notion of incidence theorems and prove several tropical versions of classical theorems like converse Pascal, Fano plane or Cayley-Bacharach.