View source for Tropical geometry

[[File:Cubique tropicale min.svg|thumbnail|A tropical cubic curve]]
'''Tropical geometry''' is a relatively new area in [[Mathematics|mathematics]], which might loosely be described as a [[Piecewise linear manifold|piece-wise linear]] or skeletonized version of [[Algebraic geometry|algebraic geometry]].  Its leading ideas had appeared in different forms in the earlier works of [[Biography:George Bergman|George M. Bergman]] and of Robert Bieri and John Groves, but 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 geometry|enumerative algebraic geometry]] found by Grigory Mikhalkin.

== History of the name ==
The adjective ''tropical'' in the name of the area was coined by French mathematicians in honor of the Hungarian-born {{wipe|Brazil}}ian mathematician [[Biography:Imre Simon|Imre Simon]], who wrote on the field. Jean-Eric Pin<ref name=Pin1998>Jean-Eric Pin. Tropical semirings. Idempotency (Bristol, 1994). Publ. Newton Inst 11 (1998), pp. 50–69.</ref> attributes the coinage to Dominique Perrin, whereas Simon<ref name=Simon1988>Imre Simon. Recognizable sets with multiplicities in the tropical semiring. Mathematical Foundations of Computer Science (1988), pp. 107–120.</ref> himself attributes the word to Christian Choffrut.

== Basic definitions ==

The following definitions use the ''min'' convention'', ''that tropical addition is classical minimum''.  ''It is also possible to cast the whole subject in terms of  the ''max'' convention, negating throughout, and several authors make this choice''.'' The basic ideas of tropical analysis have been developed independently in the same notations by mathematicians working in various fields (see <ref>{{Cite journal|last=Cuninghame-Green|first=Raymond  A.| year=1979|title=Minimax algebra|url=|journal=Lecture Notes in Economics and Mathematical Sciences|volume=166|pages=|publisher=Springer  |isbn=978-3-540-09113-4}}</ref> and references therein). In 1987 Victor Pavlovich Maslov  introduced a tropical version of integration procedure. He also noticed that the [[Legendre transform]]ation and solutions of the [[Physics:Hamilton–Jacobi equation|Hamilton–Jacobi equation]] are linear operations in the tropical sense.<ref>{{Cite journal|last=Maslov|first=Victor|year=1987|title=On a new superposition principle for optimization problems|url=|journal=Russian Mathematical Surveys|volume=42:3|pages=43–54|via=United Kingdom, IOP Publishing Ltd.|bibcode=1987RuMaS..42...43M|doi=10.1070/RM1987v042n03ABEH001439}}</ref>

The ''tropical semiring'' (also known as a ''tropical algebra''<ref name=Litvinov2009>{{cite book|last1=Litvinov|first1=Grigoriĭ Lazarevich|last2=Sergeev|first2=Sergej Nikolaevič|title=Tropical and Idempotent Mathematics: International Workshop Tropical-07, Tropical and Idempotent Mathematics|date=2009|publisher=American Mathematical Society| isbn=9780821847824|page=8|url=http://www.mccme.ru/tropical12/Tropics2012final.pdf|accessdate=15 September 2014}}</ref> or, with the max convention, the ''[[Max-plus algebra|max-plus algebra]]'', due to the name of its operations) is a [[Semiring|semiring]] (ℝ ∪ {∞}, ⊕, ⊗), with the operations as follows:
: <math>x \oplus y = \min\{x, y \},</math>
: <math>x \otimes y = x + y.</math>

Tropical exponentiation is defined in the usual way as iterated tropical products (see [[Exponentiation#In abstract algebra|exponentiation#In abstract algebra]]).

A [[Monomial|monomial]] of variables in this semiring is a [[Linear map|linear map]], represented in classical arithmetic as a linear function of the variables with integer coefficients.<ref name=SpeyerSturmfels2009>{{citation|first1=David|last1=Speyer |first2=Bernd|last2=Sturmfels|title=Tropical mathematics|journal=Mathematics Magazine|volume=82|issue=3|year=2009|pages=163–173|url=https://math.berkeley.edu/~bernd/mathmag.pdf}}</ref>  A [[Polynomial|polynomial]] in the semiring is the minimum of a finite number of such monomials, and is therefore a [[Concave function|concave]], [[Continuous function|continuous]], [[Piecewise linear function|piecewise linear]] function.

The set of points where a tropical polynomial ''F'' is non-differentiable is called its associated '''tropical hypersurface'''.

There are two important characterizations of these objects:

# Tropical hypersurfaces are exactly the rational polyhedral complexes satisfying a "zero-tension" condition.<ref name=SpeyerSturmfels2009/>
# Tropical surfaces are exactly the non-Archimedean [[Amoeba (mathematics)|amoebas]] over an [[Algebraically closed field|algebraically closed]] non-Archimedean field ''K''.<ref>{{cite book | last=Mikhalkin | first=Grigory | chapter=Amoebas of algebraic varieties and tropical geometry | editor1-last=Donaldson | editor1-first=Simon | editor1-link=Simon Donaldson | editor2-first=Yakov | editor2-last=Eliashberg | editor2-link=Yakov Eliashberg | editor3-first=Mikhael | editor3-last=Gromov | editor3-link=Mikhail Leonidovich Gromov | title=Different faces of geometry | location=New York, NY | publisher=Kluwer Academic/Plenum Publishers | series=International Mathematical Series (New York) | volume=3 | pages=257–300 | year=2004 | isbn=0-306-48657-1 | zbl=1072.14013 }}</ref>

These two characterizations provide a "dictionary" between combinatorics and algebra. Such a dictionary can be used to take an algebraic problem and solve its easier combinatorial counterpart instead.

The tropical hypersurface can be generalized to a '''tropical variety''' by taking the non-Archimedean amoeba of ideals ''I'' in <math>K[x_1,\ldots, x_n]</math> instead of polynomials. The tropical variety of an ideal ''I'' equals the intersection of the tropical hypersurfaces associated to every polynomial in ''I''. This intersection can be chosen to be finite.

There are a number of articles and surveys on tropical geometry.{{example needed}} The study of '''tropical curves''' (tropical hypersurfaces in <math>\mathbb{R}^2)</math> is particularly well developed. In fact, for this setting, mathematicians have established analogues of many classical theorems; e.g., [[Pappus's hexagon theorem]], [[Bézout's theorem]], the degree-genus formula, and the [[Elliptic curve|group law of the cubics]]<ref>{{cite book | last1=Chan | first1=Melody | last2=Sturmfels | first2=Bernd |  chapter=Elliptic curves in honeycomb form | zbl=1312.14142 | editor1-last=Brugallé | editor1-first=Erwan | title=Algebraic and combinatorial aspects of tropical geometry. Proceedings based on the CIEM workshop on tropical geometry, International Centre for Mathematical Meetings (CIEM), Castro Urdiales, Spain, December 12–16, 2011 | location=Providence, RI | publisher=American Mathematical Society | isbn=978-0-8218-9146-9 | series=Contemporary Mathematics | volume=589 | pages=87–107 | year=2013 | arxiv=1203.2356 | bibcode=2012arXiv1203.2356C }}</ref> all have tropical counterparts.

== Applications ==
 
A tropical line appeared in Paul Klemperer's design of [[Finance:Auction|auction]]s used by the Bank of England during the financial crisis in 2007.<ref>{{Cite web|url = http://www.economics.ox.ac.uk/General-News/how-geometry-came-to-the-rescue-during-the-banking-crisis-video|title = How geometry came to the rescue during the banking crisis|accessdate = 24 March 2014|website = Department of Economics, University of Oxford}}</ref> Yuichi 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.<ref>Shiozawa, Yuichi, International trade theory and exotic algebra, ''Evolutionary and Institutional Economcis Review'', '''12''': 177-212. June, 2015. This is a digest of Y. Shiozawa, "Subtropical Convex Geometry as the Ricardian Theory of International Trade," draft paper in his ResearchGate page.</ref>
 
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.<ref>{{cite arXiv |last= Krivulin |first= Nikolai |eprint=1408.0313v1 |title= Tropical optimization problems |class=math.OC |year=2014 }}</ref> A tropical counterpart of [[Abel–Jacobi map]] can be applied to a crystal design.<ref>[[Biography:Toshikazu Sunada|Sunada T.]] (2012), ''Topological Crystallography ---With a View Towards Discrete Geometric Analysis---'', Surveys and Tutorials in the Applied Mathematical Sciences, Vol. 6, Springer</ref> The weights in a weighted finite-state transducer are often required to be a tropical semiring. Tropical geometry shows [[Physics:Self-organized criticality|Self-organized criticality]] behaviour <ref>{{Cite journal|last=Kalinin|first=N.|last2=Guzmán-Sáenz|first2=A.|last3=Prieto|first3=Y.|last4=Shkolnikov|first4=M.|last5=Kalinina|first5=V.|last6=Lupercio|first6=E.|date=2018-08-15|title=Self-organized criticality and pattern emergence through the lens of tropical geometry|url=http://www.pnas.org/content/115/35/E8135|journal=Proceedings of the National Academy of Sciences|language=en|pages=201805847|doi=10.1073/pnas.1805847115|issn=0027-8424|pmid=30111541}}</ref>. 

==See also==
* [[Max-plus algebra]]
* [[Tropical compactification]]
* [[Tropical cryptography]]

==Notes==
{{reflist}}

==References==
{{Refbegin}}
* Maslov, Victor (1986). "New superposition principle for optimization problems", Séminaire sur les Équations aux Dérivées Partielles 1985/6, Centre de Mathématiques de l’École Polytechnique, Palaiseau, exposé 24.
* Maslov, Victor (1987). "Méthodes Opératorielles".  Moscou, Mir, 707 p. (See Chapter 8, Théorie linéaire sur semi moduli, pp.&nbsp;652–701).
*{{cite journal |last=Bogart |first=Tristram |last2=Jensen |first2=Anders |last3=Speyer |first3=David |last4=Sturmfels |first4=Bernd | author4-link=Bernd Sturmfels | last5=Thomas |first5=Rekha |arxiv=math/0507563v1 |title=Computing Tropical Varieties |year=2005 |doi=10.1016/j.jsc.2006.02.004 |volume=42 |journal=Journal of Symbolic Computation |pages=54–73}}
*{{cite arXiv |last=Einsiedler |first=Manfred |last2=Kapranov |first2=Mikhail |last3=Lind |first3=Douglas |eprint=math/0408311v2 |title=Non-archimedean amoebas and tropical varieties |year=2005  }}
*{{cite arXiv |last= Gathmann |first= Andreas |eprint=math/0601322v1 |title=Tropical algebraic geometry |year= 2006 }}
*{{cite book|last=Gross|first=Mark|title=Tropical geometry and mirror symmetry|year=2010|publisher=Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society with support from the National Science Foundation|location=Providence, R.I.|isbn=9780821852323}}
*{{cite book|last=Itenberg|first=Illia|author2=Grigory Mikhalkin |author3=Eugenii Shustin  |title=Tropical algebraic geometry|year=2009|publisher=Birkhäuser Basel|location=Basel|isbn=9783034600484|edition=2nd | zbl=1165.14002 }}
*{{cite book|last=Maclagan|first=Diane|last2=Sturmfels|first2=Bernd |title=Introduction to tropical geometry|year=2015|publisher=American Mathematical Soc.|isbn=9780821851982 }}
*{{cite arXiv |last= Mikhalkin |first=Grigory |eprint=math/0601041v2 |title=Tropical Geometry and its applications |year=2006  }}
*{{cite arXiv |last= Mikhalkin |first=Grigory |eprint=math/0312530v4 |title=Enumerative tropical algebraic geometry in R2 |year=2004 }}
*{{cite arXiv |last= Mikhalkin |first=Grigory |eprint=math/0403015v1 |title=Amoebas of algebraic varieties and tropical geometry |year=2004  }}
*{{cite journal | last=Pachter| first=Lior |  last2=Sturmfels | first2=Bernd |  title=Tropical geometry of statistical models|journal=Proceedings of the National Academy of Sciences of the United States of America | year=2004 | volume=101 | issue=46 | pages=16132–16137 | doi=10.1073/pnas.0406010101 | zbl=1135.62302| arxiv=q-bio/0311009 | bibcode=2004PNAS..10116132P }}
*{{cite arXiv |last= Speyer |first=David E. |eprint=math/0304218v3 |title=The Tropical Grassmannian |year=2003 }}
*{{cite journal |last=Speyer |first=David |last2=Sturmfels |first2=Bernd |  arxiv=math/0408099 |title=Tropical Mathematics | origyear=2004 | year= 2009 | zbl=1227.14051 | journal=Mathematics Magazine | volume=82 | number=3 | pages=163–173 | doi=10.4169/193009809x468760}}
*{{cite arXiv |last= Theobald |first= Thorsten |eprint=math/0306366v2 |title=First steps in tropical geometry |year=2003  }}
{{Refend}}

==Further reading==
* {{cite book | editor1-last=Amini | editor1-first=Omid | editor2-last=Baker | editor2-first=Matthew | editor3-last=Faber | editor3-first=Xander | title=Tropical and non-Archimedean geometry. Bellairs workshop in number theory, tropical and non-Archimedean geometry, Bellairs Research Institute, Holetown, Barbados, USA, May 6–13, 2011 | zbl=1281.14002 | series=Contemporary Mathematics | volume=605 | <!-- subseries=Centre de Recherches Mathématiques Proceedings | --> location=Providence, RI | publisher=American Mathematical Society | isbn=978-1-4704-1021-6 | year=2013 }}

==External links==
* [https://archive.org/movies/details-db.php?id=4603 Tropical Geometry, I]

[[Category:Algebraic geometry]]

{{Sourceattribution|Tropical geometry|1}}