Tropical Geometry

The main topic of this book is the tropical semiring. Tropical geometry is a relatively new, but rapidly growing field in mathematics. The tropical semiring first appeared in computer sciences, but has deep relations with many fields of mathematics. These include algebraic geometry, mirror symmetry, complex analysis, combinatorics, logic, computer algebra and statistical models. In this book the main topic is the relation between tropical geometry and Voronoi diagrams. The author starts by discussing tropical addition and multiplication and what the graphs of polynomials, defined using this type of addition and multiplication, look like. Next, she discusses the notion of a tropical amoeba and how Voronoi diagrams relate to this notion. She will also consider power diagrams, a type of generalization of the Voronoi diagram, and look at their relation to tropical amoebas. She will show that every tropical polynomial defines a power diagram and, even better, also a Voronoi diagram. Lastly the author also approaches tropical geometry from the view of algebraic geometry. She will show how the spine of an amoeba in algebraic geometry relates to the tropical amoeba.