Tropical geometry

Summary

Tropical geometry studies polynomials over the tropical semiring, where addition is replaced by minimum (or maximum) and multiplication by ordinary addition, see max-plus algebra. Tropical polynomials are piecewise linear functions, and their zero sets, the tropical varieties, are polyhedral complexes. Through a valuation, an algebraic variety degenerates to a tropical variety that keeps much of its combinatorial information, such as degrees and intersection numbers. This makes problems of enumerative geometry and of toric varieties accessible to combinatorics.