In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
would become
. 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
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
in which polynomial graphs resemble
piecewise linear meshes, and in which numbers belong to the
tropical semiring
In idempotent analysis, the tropical semiring is a semiring of extended real numbers with the operations of minimum (or maximum) and addition replacing the usual ("classical") operations of addition and multiplication, respectively.
The tropical s ...
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 were developed independently using the same notation by mathematicians working in various fields. The central ideas of tropical geometry appeared in different forms in a number of earlier works. For example,
Victor Pavlovich Maslov
Viktor Pavlovich Maslov (russian: Виктор Павлович Маслов; born 15 June 1930, in Moscow) is a Russian mathematical physicist. He is a member of the Russian Academy of Sciences. He obtained his doctorate in physico-mathematical ...
introduced a tropical version of the process of integration. He also noticed that the
Legendre transformation
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions of ...
and solutions of the
Hamilton–Jacobi equation
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechan ...
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 was motivated by its application to
enumerative algebraic geometry, with ideas from
Maxim Kontsevich
Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques an ...
and works by Grigory Mikhalkin among others.
The adjective ''
tropical
The tropics are the regions of Earth surrounding the Equator. They are defined in latitude by the Tropic of Cancer in the Northern Hemisphere at N and the Tropic of Capricorn in
the Southern Hemisphere at S. The tropics are also referred to ...
'' was coined by French mathematicians in honor of the
Hungarian-born
Brazil
Brazil ( pt, Brasil; ), officially the Federative Republic of Brazil (Portuguese: ), is the largest country in both South America and Latin America. At and with over 217 million people, Brazil is the world's fifth-largest country by area ...
ian computer scientist
Imre Simon
Imre Simon (August 14, 1943 – August 13, 2009) was a Hungarian-born Brazilian mathematician and computer scientist.
His research mainly focused on theoretical computer science, automata theory, and tropical mathematics, a subject he founded, ...
, who wrote on the field.
Jean-Éric Pin
Jean-Éric Pin is a French mathematician and theoretical computer scientist known for his contributions to the algebraic automata theory and semigroup theory. He is a CNRS research director.
Biography
Pin earned his undergraduate degree from ENS ...
attributes the coinage to
Dominique Perrin
Dominique Pierre Perrin (b. 1946) is a French mathematician and Theoretical computer science, theoretical computer scientist known for his contributions to coding theory and to combinatorics on words. He is a professor of the University of Marne-la ...
,
whereas Simon himself attributes the word to Christian Choffrut.
Algebra background
Tropical geometry is based on the
tropical semiring
In idempotent analysis, the tropical semiring is a semiring of extended real numbers with the operations of minimum (or maximum) and addition replacing the usual ("classical") operations of addition and multiplication, respectively.
The tropical s ...
. This is defined in two ways, depending on max or min convention.
The ''min tropical semiring'' is the
semiring
In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse.
The term rig is also used occasionally—this originated as a joke, suggesting that rigs ar ...
, with the operations:
:
:
The operations
and
are referred to as ''tropical addition'' and ''tropical multiplication'' respectively. The
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
for
is
, and the identity element for
is 0.
Similarly, the ''max tropical semiring'' is the semiring
, with operations:
:
:
The identity element for
is
, and the identity element for
is 0.
These semirings are isomorphic, under negation
, 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
Value or values may refer to:
Ethics and social
* Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them
** Values (Western philosophy) expands the notion of value beyo ...
.
Some common valuated fields encountered in tropical geometry (with min convention) are:
*
or
with the trivial valuation,
for all
.
*
or its extensions with the
p-adic valuation
In number theory, the valuation or -adic order of an integer is the exponent of the highest power of the prime number that divides .
It is denoted \nu_p(n).
Equivalently, \nu_p(n) is the exponent to which p appears in the prime factorization of ...
,
for ''a'' and ''b'' coprime to ''p''.
* The field of
Laurent series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
(integer powers), or the field of (complex)
Puiseux series
In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series
: \begin
x^ &+ 2x^ + x^ + 2x^ + x^ + x^5 + \cdots\\
&=x^+ 2x^ + x^ + 2x^ + x^ + ...
, with valuation returning the smallest exponent of ''t'' appearing in the series.
Tropical polynomials
A ''tropical polynomial'' is a function
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
Concave or concavity may refer to:
Science and technology
* Concave lens
* Concave mirror
Mathematics
* Concave function, the negative of a convex function
* Concave polygon, a polygon which is not convex
* Concave set
* The concavity
In ca ...
,
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
, and
piecewise linear.
:
Given a polynomial ''f'' in the
Laurent polynomial ring