Tropical geometry

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Tropical geometry is a relatively new area in mathematics. It has appeared in different guises in previous works of Bergman and of Bieri and Groves, but only since the late nineties an effort has 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. Being such a recent area there is not a standard formulation for the theory. Some definitions aren't universal, 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). Besides that, it has no deeper meaning.

1 Basic definitions
2 External links
- 2.1 Introductory articles and surveys
- 2.2 Talk on tropical geometry

[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 application, and a polynomial is the minimum of a finite number of such functions, and therefore a concave, piecewise linear function.

We define the tropical hypersurface associated to a tropical polynomial F as the set of points where F is non-differentiable.

There are two important characterizations of this 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 was 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, we can prove analogues to all the classic theorems: Pappus's theorem, Bézout's theorem, degree-genus formula, group law of the cubics; all of them have tropical counterparts.