Tropical geometry

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:

${\displaystyle x\oplus y=\min\{x,y\},}$

${\displaystyle x\otimes y=x+y.}$

So for example, the classical polynomial ${\displaystyle x^{3}+2xy+y^{4}}$ would become ${\displaystyle \min\{x+x+x,\;2+x+y,\;y+y+y+y\}}$. 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 geometry, such as the Brill–Noether theorem, using the tools of tropical geometry.^[1]