Tropical geometry

I will explain how to do computations in McMullen's polytope algebra and Chow cohomology of toric varieties, using tropical geometry.

Tropical geometry is the geometry over the tropical semiring, which is the set of real numbers where the tropical addition is taking the minimum, and the tropical multiplication is the ordinary addition. As the ordinary linear and polynomial algebra give rise to convex geometry and algebraic geometry, tropical linear and polynomial algebra give rise to tropical convex geometry and tropical algebraic geometry.

The tropical variety of a polynomial ideal I in n variables over Q is a polyhedral complex in n-dimensional space. We may consider it as a subfan of the Groebner fan of I. The polyhedral cones in the Groebner fan can be computed using Groebner bases and by applying Groebner walk techniques. This gives one method for computing the tropical variety of I. We show how the method can be refined by applying a connectivity result for tropical varieties of prime ideals and an algorithm for constructing tropical bases of curves. The presented algorithms have