First steps in tropical geometry

TL;DR

This paper introduces tropical algebraic geometry, focusing on plane curves and linear spaces, and presents new results including classifications of quadrics and a counterexample to a classical theorem.

Contribution

It provides an introductory overview of tropical geometry with novel findings on quadrics and incidence theorems in the tropical setting.

Findings

Complete description of quadrics through four points in tropical projective plane

Counterexample to the incidence version of Pappus' Theorem

Emphasis on polyhedral complexes resembling algebraic varieties

Abstract

Tropical algebraic geometry is the geometry of the tropical semiring $(\mathbb{R},\min,+)$. Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on plane curves and linear spaces. New results include a complete description of the families of quadrics through four points in the tropical projective plane and a counterexample to the incidence version of Pappus' Theorem.