Tropical geometry

Tropical geometry is a relatively new area in mathematics, which might loosely be described as a piece-wise linear or skeletonized version of algebraic geometry, using the tropical semiring instead of a field. 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 classic results from algebraic geometry, such as the Brill–Noether theorem, using the tools of tropical geometry. The basic ideas of tropical analysis have been developed independently in the same notations by mathematicians working in various fields (see and references therein). The leading ideas of tropical geometry had appeared in different forms in the 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 has been motivated by the applications to enumerative algebraic geometry, with ideas from Maxim Kontsevich and works by Grigory Mikhalkin among others. The adjective tropical in the name of the area was coined by French mathematicians in honor of the Hungarian-born Brazilian computer scientist Imre Simon, who wrote on the field. Jean-Eric Pin attributes the coinage to Dominique Perrin, whereas Simon himself attributes the word to Christian Choffrut. 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 (ℝ ∪ {+∞}, ⊕, ⊗), with the operations: The operations ⊕ and ⊗ are referred to as tropical addition and tropical multiplication respectively. The unit for ⊕ is +∞, and the unit for ⊗ is 0. Similarly, the max tropical semiring is the semiring (ℝ ∪ {−∞}, ⊕, ⊗), with operations: