Tropical geometry

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In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition: x⊕y=min⁡{x,y}, x⊗y=x+y. So for example, the classical polynomial x3+xy+y4 would become min⁡{x+x+x,x+y,y+y+y+y}. Such polynomials and their solutions have important applications in optimization problems, for example, the problem of optimizing departure times for a network of trains. Tropical geometry is a variant of algebraic geometry in which polynomial graphs resemble piecewise linear meshes, and in which numbers belong to the tropical semiring instead of a field. Because classical and tropical geometry are closely related, results and methods can be converted between them. Algebraic varieties can be mapped to a tropical counterpart and, since this process still retains some geometric information about the original variety, it can be used to help prove and generalize classical results from algebraic...

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