Tropical geometry

TL;DR

Tropical geometry is a mathematical framework that redefines arithmetic operations to study piecewise-linear geometric objects called tropical varieties, linking them to classical algebraic geometry.

Contribution

The paper introduces tropical algebraic structures, explores tropical varieties in low dimensions, and discusses their relationship with traditional algebraic geometry.

Findings

Tropical polynomials define piecewise-linear varieties.

Tools for analyzing tropical varieties in 2D and 3D are developed.

Connections between tropical and classical algebraic geometry are examined.

Abstract

Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build with these new operations. These equations define piecewise-linear geometric objects called tropical varieties. We explore these tropical varieties in two and three dimensions, building up discrete tools for studying them and determining their geometric properties. We then discuss the relationship between tropical geometry and algebraic geometry, which considers shapes defined by usual polynomial equations.