Tropical variety

Summary

A tropical variety is a polyhedral complex in $\R^n$ attached to an algebraic subvariety of a torus over a field with a valuation. It is the set where tropical polynomials are "zero", meaning that the minimum of their linear terms is attained at least twice. Over an algebraically closed field with a nontrivial valuation, it equals the closure of the image of the variety under the coordinatewise valuation. This is the fundamental theorem of tropical geometry. Tropical varieties keep dimension, degree and intersection data of the original variety in combinatorial form.