Tropical Geometry

Tropical geometry still is a rather young field of mathematics which nevertheless over the past couple of years has proved to be rather powerful for tackling questions in enumerative geometry (see e.g. [Mik05], [GM08], [IKS04]). It also has been used to provide alternative algorithmic means to eliminate variables (see [StY08]). People from applied mathematics such as optimisation or control theory are interested in it as a kind of algebraic geometry over the max-plus algebra (see e.g. [CGQ99]). Even though all these people talk about tropical varieties in some way or the other, their ideas of what a tropical variety should be differ quite a bit. For enumerative questions it is most helpful to consider tropical curves as parametrised objects, while for elimination one has to take an implicit point of view, and for many other questions a purely combinatorial description seems best. In general the classes of objects considered do not completely coincide, but they share a sufficiently large overlap. For the purpose of this paper we will mainly consider the implicit approach. We will explain some of the combinatorial structure which is inherent in all the different approaches, and we will focus on certain computational questions in tropical geometry. One could think of tropical geometry as being a shadow of classical algebraic geometry, which carries enough information to shed some light on the classical objects but which at the same time is light enough to be easier to deal with, or better to allow the application of tools from other areas of mathematics. The base field over which the classical objects live should be algebraically closed and carry a non-trivial nonarchimedean valuation into the real numbers. The prototype