What is tropical geometry about?

What is tropical geometry about?

  junio 22, 2017    Danilo Jose Polo Ojito

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Tropical geometry is an exciting new field at the interface between algebraic geometry and combinatorics with connections to many other fields. At its heart it is geometry over the tropical semiring, which is R∪{∞} with the usual operations of addition and multiplication replaced by minimum and addition respectively. This turns polynomials into piecewise linear functions,and replaces an algebraic variety by an object from polyhedral geometry,which can be regarded as a “combinatorial shadow” of the original variety.



In tropical algebra, the sum of two numbers is their minimum and the prod-uct of two numbers is their sum. This algebraic structure is known as thetropical semiring or as the min-plus algebra. With minimum replaced by maximum we get the isomorphic max-plus algebra. The adjective “tropical”was coined by French mathematicians, notably Jean Eric Pin, to honor their Brazilian colleague Imre Simon , who pioneered the use of min-plus algebra in optimization theory. There is no deeper meaning to the adjective “tropical”. It simply stands for the French view of Brazil.
The origins of algebraic geometry lie in the study of zero sets of systems of multivariate polynomials. These objects are algebraic varieties, and they include familiar examples such as plane curves and surfaces in three dimensional space. It makes perfect sense to define polynomials and rational functions over the tropical semiring. These functions are piecewise linear.Algebraic varieties can also be defined in the tropical setting.  They are now subsets of Rn that are composed of convex polyhedra. Thus tropical algebraic geometry is a piecewise-linear version of algebraic geometry.

Álgebra tropical

Consider the set of real numbers $\mathbb{R}$ endowed with the following operations:

                                $x\oplus y=min\{x,y\}$      $x\odot y=x+y$

Ejemplo:

$3\oplus 4=min\{3,4\}=3$               $3\odot 4= 3+4 =7$

For all $x,y\in \mathbb{R}$ it is easy to verify that these operations are commutative, associative, and $\odot$ is distributive with respect to $\oplus$.

The neutral element in the operation $\odot$ es el $0$, as $x\odot 0=x+0=x$, but the operation $\oplus$ lacks neutral element in $\mathbb{R}$ $\mathbb{R}_{min}=\mathbb{R}\cup\{+\infty\}$ y se verfica que  $x\oplus +\infty=min\{x,+\infty\}=x$, therefore the neutral element for $\oplus$ is $+\infty$.

But they wonder why it is called  tropical semiring and not tropical ring, the answer is that the operation $\oplus$ is not invertive, since for example equation $3\oplus x=+\infty$ no solution for any $x\in \mathbb{R}$, as $min\{3,x\}$ it will not be equal to $+\infty$.

So the triplet $(\mathbb{R}_{min},\oplus,\odot)$ is called semiring tropical.

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