Tropical geometry

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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 guises in previous works of Bergman and of Bieri and Groves, but only since the late nineties has an effort been made to consolidate the basic definitions of the theory. This effort has been in great part motivated by the strong applications to enumerative algebraic geometry uncovered by Grigory Mikhalkin. Because it is such a recent and evolving area of research, there is not a standard formulation for the theory: some definitions are not universally accepted, and some basic results lack proofs.

The adjective tropical is given in honor of the Brazilian mathematician Imre Simon, who pioneered the field. It simply reflects the French view on Brazil (as it was coined by a Frenchman). Beyond that, it has no deeper meaning.

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[edit] Basic definitions

Consider the tropical semiring (also known as the min-plus algebra due to the definition of the semiring). This semiring, (R ∪ {∞}, ⊕, ⊗), is defined with the operations as follows:

x \oplus y = \min\{\, x, y \,\},\,
x \otimes y = x + y.\,

A monomial in this semiring is simply a linear map, and a polynomial is the minimum of a finite number of such functions, and therefore a concave, piecewise linear function.

The set of points where a tropical polynomial F is non-differentiable is called its associated tropical hypersurface.

There are two important characterizations of these objects:

1. Tropical hypersurfaces are exactly the rational polyhedral complexes verifying a "zero-tension" condition.

2. Tropical surfaces are exactly the non-archimedean amoebas over an algebraically closed field K with a non-archimedean valuation.

These two characterizations provide a "dictionary" between combinatorics and algebra. Such a dictionary can be used to take an algebraic problem and solve its easier combinatorial counterpart instead.

The tropical hypersurface can be generalized to a tropical variety by taking the non-archimedean amoeba of ideals I in K[x_1,\dots,x_n] instead of polynomials. It has been proved that the tropical variety of an ideal I equals the intersection of the tropical hypersurfaces associated to every polynomial in I. This intersection can be chosen to be finite.

There are a number of articles and surveys on tropical geometry. The study of tropical curves (tropical hypersurfaces in \mathbb{R}^2) is particularly well studied. In fact, for this setting, mathematicians have established analogues of many classical theorems; e.g., Pappus's theorem, Bézout's theorem, the degree-genus formula, and the group law of the cubics all have tropical counterparts.

[edit] External links

[edit] Introductory articles and surveys

[edit] Talk on tropical geometry