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. Its leading ideas had appeared in different forms in the earlier works of George M. Bergman and of Robert Bieri and John Groves, but 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 found by Grigory Mikhalkin.

The adjective tropical in the name of the area was coined by French mathematicians in honor of the Hungarian-born Brazilian mathematician 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.

We will use the min convention, that tropical addition is classical minimum. It is also possible to cast the whole subject in terms of the max convention, negating throughout, and several authors make this choice. The basic ideas of tropical analysis have been developed independently in the same notations by mathematicians working in various fields (see and references therein). In 1987 V. P. Maslov introduced a tropical version of integration procedure. He also noticed that the Legendre transformation and solutions of the Hamilton-Jacobi equation are linear operations in the tropical sense.

The tropical semiring (also known as a tropical algebra or, with the max convention, the max-plus algebra, due to the name of its operations) is a semiring (ℝ ∪ {∞}, ⊕, ⊗), with the operations as follows:

Tropical exponentiation is defined in the usual way as iterated tropical products (see exponentiation#In abstract algebra).

A monomial of variables in this semiring is a linear map, represented in classical arithmetic as a linear function of the variables with integer coefficients. A polynomial in the semiring is the minimum of a finite number of such monomials, and is therefore a concave, continuous, piecewise linear function.

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