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
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 or computing Gromov Witten invariants, using the tools of tropical geometry.
[1] The basic ideas of tropical analysis were developed independently using the same notation by mathematicians working in various fields.
[2] The central ideas of tropical geometry appeared in different forms in a number of 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.
[3] However, only since the late 1990s has an effort been made to consolidate the basic definitions of the theory.
This was motivated by its application to enumerative algebraic geometry, with ideas from Maxim Kontsevich[4] and works by Grigory Mikhalkin[5] among others.
The adjective tropical was coined by French mathematicians in honor of the Hungarian-born Brazilian computer scientist Imre Simon, who wrote on the field.
This is defined in two ways, depending on max or min convention.
The tropical semiring operations model how valuations behave under addition and multiplication in a valued field.
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
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.
The equivalence of these definitions is referred to as the Fundamental Theorem of Tropical Geometry.
is a principal ideal generated by a Laurent polynomial f, and the tropical variety
to be Then define Since we are working in the Laurent ring, this is the same as the set of weight vectors for which
By acting coordinate-wise, v defines a map from the algebraic torus
, then the above definition can be adapted by extending scalars to larger field which does have a dense valuation.
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.
[13] 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.
A tropical line appeared in Paul Klemperer's design of auctions used by the Bank of England during the financial crisis in 2007.
[17] Yoshinori Shiozawa defined subtropical algebra as max-times or min-times semiring (instead of max-plus and min-plus).
[18][non-primary source needed] Tropical geometry has also been used for analyzing the complexity of feedforward neural networks with ReLU activation.
[19] 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.
[22] Tropical geometry has also found applications in several topics within theoretical high energy physics.
In particular, tropical geometry has been used to drastically simplify string theory amplitudes to their field-theoretical limits [23] and has found connections to constructions such as the Amplituhedron[24] and tropological (topological Carrollian) sigma models.
