Tropical Geometry

Abstract

The main topic of this thesis is the tropical semiring. We discuss tropical addition and multiplication and what the graphs of polynomials, defined using this type of addition and multiplication, look like. We will discuss the notion of a tropical amoeba and how Voronoi diagrams relate to this notion. We will also consider power diagrams, a type of generalization of the Voronoi diagram, and look at their relation to tropical amoebas. We will show that every tropical polynomial defines a power diagram and, even better, also a Voronoi diagram. We will also approach tropical geometry from the view of algebraic geometry. We discuss the notion of an amoeba in algebraic geometry and consider the spine of this amoeba. Finally we will show how the spine of an amoeba relates to the tropical amoeba