Tropical geometry

In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition:
So for example, the classical polynomial formula_3 would become formula_4. Such polynomials and their solutions have important applications in optimization problems, for example the problem of optimizing departure times for a network of trains.
Tropical geometry is a variant of algebraic geometry in which polynomial graphs resemble piecewise linear meshes, and in which numbers belong to the tropical semiring instead of a field. Because classical and tropical geometry are closely related, results and methods can be converted between them. 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 and generalize classical results from algebraic geometry, such as the Brill–Noether theorem, using the tools of tropical geometry.
History.
The basic ideas of tropical analysis have been developed independently in the same notations by mathematicians working in various fields. 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-Éric Pin attributes the coinage to Dominique Perrin, whereas Simon himself attributes the word to Christian Choffrut.
Algebra background.
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 formula_5, with the operations:
The operations formula_8 and formula_9 are referred to as "tropical addition" and "tropical multiplication" respectively. The identity element for formula_8 is formula_11, and the identity element for formula_9 is 0.
Similarly, the "max tropical semiring" is the semiring formula_13, with operations:
The neutral element for formula_8 is formula_17, and the neutral element for formula_9 is 0.
These semirings are isomorphic, under negation formula_19, and generally one of these is chosen and referred to simply as the "tropical semiring". Conventions differ between authors and subfields: some use the "min" convention, some use the "max" convention.
The tropical semiring operations model how valuations behave under addition and multiplication in a valued field.
Some common valuated fields encountered in tropical geometry (with min convention) are:
Tropical polynomials.
A "tropical polynomial" is a function formula_28 that can be expressed as the tropical sum of a finite number of "monomial terms". A monomial term is a tropical product (and/or quotient) of a constant and variables from formula_29. Thus a tropical polynomial "F" is the minimum of a finite collection of affine-linear functions in which the variables have integer coefficients, so it is concave, continuous, and piecewise linear.
Given a polynomial "f" in the Laurent polynomial ring formula_31 where "K" is a valued field, the "tropicalization" of "f", denoted formula_32, is the tropical polynomial obtained from "f" by replacing multiplication and addition by their tropical counterparts and each constant in "K" by its valuation. That is, if 
then
The set of points where a tropical polynomial "F" is non-differentiable is called its associated "tropical hypersurface", denoted formula_35 (in analogy to the vanishing set of a polynomial). Equivalently, formula_35 is the set of points where the minimum among the terms of "F" is achieved at least twice. When formula_37 for a Laurent polynomial "f", this latter characterization of formula_35 reflects the fact that at any solution to formula_39, the minimum valuation of the terms of "f" must be achieved at least twice in order for them all to cancel.
Tropical varieties.
Definitions.
For "X" an algebraic variety in the algebraic torus formula_40, the "tropical variety" of "X" or "tropicalization" of "X", denoted formula_41, is a subset of formula_42 that can be defined in several ways. The equivalence of these definitions is referred to as the "Fundamental Theorem of Tropical Geometry".
Intersection of tropical hypersurfaces.
Let formula_43 be the ideal of Laurent polynomials that vanish on "X" in formula_31. Define
When "X" is a hypersurface, its vanishing ideal formula_43 is a principal ideal generated by a Laurent polynomial "f", and the tropical variety formula_41 is precisely the tropical hypersurface formula_48.
Every tropical variety is the intersection of a finite number of tropical hypersurfaces. A finite set of polynomials formula_49 is called a "tropical basis" for "X" if formula_41 is the intersection of the tropical hypersurfaces of formula_51. In general, a generating set of formula_43 is not sufficient to form a tropical basis. The intersection of a finite number of a tropical hypersurfaces is called a "tropical prevariety" and in general is not a tropical variety.
Initial ideals.
Choosing a vector formula_53 in formula_42 defines a map from the monomial terms of formula_31 to formula_56 by sending the term "m" to formula_57. For a Laurent polynomial formula_58, define the "initial form" of "f" to be the sum of the terms formula_59 of "f" for which formula_60 is minimal. For the ideal formula_43, define its "initial ideal" with respect to formula_53 to be
Then define
Since we are working in the Laurent ring, this is the same as the set of weight vectors for which formula_65 does not contain a monomial.
When "K" has trivial valuation, formula_65 is precisely the initial ideal of formula_43 with respect to the monomial order given by a weight vector formula_53. It follows that formula_41 is a subfan of the Gröbner fan of formula_43.
Image of the valuation map.
Suppose that "X" is a variety over a field "K" with valuation "v" whose image is dense in formula_56 (for example a field of Puiseux series). By acting coordinate-wise, "v" defines a map from the algebraic torus formula_40 to formula_42. Then define
where the overline indicates the closure in the Euclidean topology. If the valuation of "K" is not dense in formula_56, then the above definition can be adapted by extending scalars to larger field which does have a dense valuation.
This definition shows that formula_41 is the non-Archimedean amoeba over an algebraically closed non-Archimedean field "K".
If "X" is a variety over formula_21, formula_41 can be considered as the limiting object of the amoeba formula_79 as the base "t" of the logarithm map goes to infinity.
Polyhedral complex.
The following characterization describes tropical varieties intrinsically without reference to algebraic varieties and tropicalization.
A set "V" in formula_42 is an irreducible tropical variety if it is the support of a weighted polyhedral complex of pure dimension "d" that satisfies the "zero-tension condition" and is connected in codimension one. When "d" is one, the zero-tension condition means that around each vertex, the weighted-sum of the out-going directions of edges equals zero. For higher dimension, sums are taken instead around each cell of dimension formula_81 after quotienting out the affine span of the cell. The property that "V" is connected in codimension one means for any two points lying on dimension "d" cells, there is a path connecting them that does not pass through any cells of dimension less than formula_81.
Tropical curves.
The study of "tropical curves" (tropical varieties of dimension one) is particularly well developed and is strongly related to graph theory. For instance, the theory of divisors of tropical curves are related to chip-firing games on graphs associated to the tropical curves.
Many classical theorems of algebraic geometry have counterparts in tropical geometry, including:
Oleg Viro used tropical curves to classify real curves of degree 7 in the plane up to isotopy. His method of "patchworking" gives a procedure to build a real curve of a given isotopy class from its tropical curve.
Applications.
A tropical line appeared in Paul Klemperer's design of auctions used by the Bank of England during the financial crisis in 2007. Yoshinori Shiozawa defined subtropical algebra as max-times or min-times semiring (instead of max-plus and min-plus). He found that Ricardian trade theory (international trade without input trade) can be interpreted as subtropical convex algebra.
Moreover, several optimization problems arising for instance in job scheduling, location analysis, transportation networks, decision making and discrete event dynamical systems can be formulated and solved in the framework of tropical geometry. A tropical counterpart of the Abel–Jacobi map can be applied to a crystal design. The weights in a weighted finite-state transducer are often required to be a tropical semiring. Tropical geometry can show self-organized criticality.