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: So for example, the classical polynomial would become . 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. History The basic ideas of tropical analysis were developed independently using the same notation by mathematicians working in various fields. The central ideas of tropical geometry appeared in different forms in a number of earlier works. For example, Victor Pavlovich Maslov introduced a tropical version of the process of integration. He also noticed that the Legendre transformation and solutions of the Hamilton–Jacobi equation are linear operations in the tropical sense. However, only since the late 1990s has an effort been made to consolidate the basic definitions of the theory. This was motivated by its application to enumerative algebraic geometry, with ideas from Maxim Kontsevich and works by Grigory Mikhalkin among others. The adjective tropical was coined by French mathematicians in honor of the Hungarian-born Brazilian computer scientist Imre Simon, who wrote on the field. Jean-Éric Pin attributes the coinage to Dominique Perrin, whereas Simon himself attributes the word to Christian Choffrut. Algebra background Tropical geometry is based on the tropical semiring. This is defined in two ways, depending on max or min convention. The min tropical semiring is the semiring , with the operations: The operations and are referred to as tropical addition and tropical multiplication respectively. The identity element for is , and the identity element for is 0. Similarly, the max tropical semiring is the semiring , with operations: The identity element for is , and the identity element for is 0. These semirings are isomorphic, under negation , and generally one of these is chosen and