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
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 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 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 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 was motivated by its application to
enumerative algebraic geometry, with ideas from
Maxim Kontsevich and works by Grigory Mikhalkin 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.
Jean-Éric Pin attributes the coinage to
Dominique Perrin
Dominique Pierre Perrin (b. 1946) is a French mathematician and 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-Vallée and currently serve ...
,
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 , with the operations:
:
:
The operations
and
are referred to as ''tropical addition'' and ''tropical multiplication'' respectively. The
identity element 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.
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,
for ''a'' and ''b'' coprime to ''p''.
* The field of
Laurent series (integer powers), or the field of (complex)
Puiseux series , 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
In geometry, a subset o ...
,
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 g ...
, and
piecewise linear.
:
Given a polynomial ''f'' in the
Laurent polynomial ring