Tropical geometry

The basic ideas of tropical analysis were developed independently using the same notation by mathematicians working in various fields.See {{Cite book|last=Cuninghame-Green|first=Raymond A.| year=1979|title=Minimax algebra|series=Lecture Notes in Economics and Mathematical Sciences|volume=166|publisher=Springer |isbn=978-3-540-09113-4 |postscript=none }} and references therein. 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.{{Cite journal|last=Maslov|first=Victor|author-link=Victor Pavlovich Maslov|year=1987|title=On a new superposition principle for optimization problems|journal=Russian Mathematical Surveys|volume=42|issue=3|pages=43–54|bibcode=1987RuMaS..42...43M|doi=10.1070/RM1987v042n03ABEH001439|s2cid=250889913 }} 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{{cite arXiv|last1=Kontsevich|first1=Maxim|author-link=Maxim Kontsevich|last2=Soibelman|first2=Yan|date=7 November 2000|title=Homological mirror symmetry and torus fibrations|eprint=math/0011041}} and works by Grigory Mikhalkin{{cite journal|last=Mikhalkin|first=Grigory|date=2005|title=Enumerative tropical algebraic geometry in R² |url=http://www.ams.org/journals/jams/2005-18-02/S0894-0347-05-00477-7/S0894-0347-05-00477-7.pdf |journal=Journal of the American Mathematical Society |volume=18 |issue=2 |pages=313–377|arxiv=math/0312530|doi=10.1090/S0894-0347-05-00477-7}} 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,{{cite book |last=Pin |first=Jean-Eric |chapter=Tropical semirings |editor-last=Gunawardena |editor-first=J. |title=Idempotency |chapter-url=https://hal.archives-ouvertes.fr/hal-00113779/file/Tropical.pdf |publisher=Cambridge University Press |series=Publications of the Newton Institute |volume=11 |year=1998 |pages=50–69 |doi=10.1017/CBO9780511662508.004 |isbn=9780511662508}} whereas Simon himself attributes the word to Christian Choffrut.{{Cite book |doi = 10.1007/BFb0017135 |volume = 324 |pages = 107–120 |series = Lecture Notes in Computer Science |year = 1988 |last1 = Simon |first1 = Imre |title = Mathematical Foundations of Computer Science 1988 |chapter = Recognizable sets with multiplicities in the tropical semiring |publisher = Springer |location = Berlin/Heidelberg |isbn = 978-3-540-50110-7}}

{{further|Tropical semiring}}

Tropical geometry is based on the tropical semiring. This is defined in two ways, depending on max or min convention.

The min tropical semiring $\backslash mathbb\{T\}$ is the semiring $\backslash mathbb\{T\}=(\backslash R\backslash cup\backslash \{+\backslash infty\backslash \},\backslash oplus,\backslash otimes)$, with the operations:

: $x\backslash oplus\; y=\backslash min\backslash \{x,y\backslash \}$,

: $x\backslash otimes\; y=x+y$.

The operations $\backslash oplus$ and $\backslash otimes$ are referred to as tropical addition and tropical multiplication respectively. The identity element for $\backslash oplus$ is $+\backslash infty$, and the identity element for $\backslash otimes$ is 0.

Similarly, the max tropical semiring $\backslash mathbb\{T\}$ is the semiring $\backslash mathbb\{T\}=(\backslash R\backslash cup\backslash \{-\backslash infty\backslash \},\backslash oplus,\backslash otimes)$, with operations:

: $x\backslash oplus\; y=\backslash max\backslash \{x,y\backslash \}$,

: $x\backslash otimes\; y=x+y$.

The identity element for $\backslash oplus$ is $-\backslash infty$, and the identity element for $\backslash otimes$ is 0.

These semirings are isomorphic, under negation $x\; \backslash mapsto\; -x$, 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 valued fields encountered in tropical geometry (with min convention) are:

• $\backslash Q$ or $\backslash Complex$ with the trivial valuation, $v(a)=0$ for all $a\backslash ne0$.
• $\backslash Q$ or its extensions with the p-adic valuation, $v\_p(p^n\; a/b)=n$ for a and b coprime to p.
• The field of Laurent series $\backslash Complex(\backslash !(t)\backslash !)$ (integer powers), or the field of (complex) Puiseux series $\backslash Complex\backslash \{\backslash !\backslash \{t\backslash \}\backslash !\backslash \}$, with valuation returning the smallest exponent of t appearing in the series.

A tropical polynomial is a function $F:\backslash R^n\backslash to\backslash R$ 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 $X\_1,\backslash ldots,X\_n$. 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.{{citation|first1=David|last1=Speyer |first2=Bernd|last2=Sturmfels|author2-link= Bernd Sturmfels |title=Tropical mathematics|journal=Mathematics Magazine|volume=82|issue=3|year=2009|pages=163–173|url=https://math.berkeley.edu/~bernd/mathmag.pdf|doi=10.1080/0025570X.2009.11953615 |s2cid=15278805 }}

:$$

\begin{align}F(X_1,\ldots,X_n)&=\left(C_1\otimes X_1^{\otimes a_{11}}\otimes\cdots\otimes X_n^{\otimes a_{n1}}\right)\oplus\cdots\oplus\left(C_s\otimes X_1^{\otimes a_{1s}}\otimes\cdots\otimes X_n^{\otimes a_{ns}}\right)\\&=\min\{C_1+a_{11}X_1+\cdots+a_{n1}X_n,\;\ldots,\;C_s+a_{1s}X_1+\cdots+a_{ns}X_n\}\end{align}

Given a polynomial $f$ in the Laurent polynomial ring $K[x\_1^\{\backslash pm\; 1\},\backslash ldots,x\_n^\{\backslash pm1\}]$ where $K$ is a valued field, the tropicalization of $f$, denoted $\backslash operatorname\{Trop\}(f)$, 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

:$f=\backslash sum\_\{i=1\}^s\; c\_i\; x^\{A\_i\}\backslash quad\; \backslash text\{with\; \}A\_1,\backslash ldots,A\_s\backslash in\backslash Z^n$,

then

:$\backslash operatorname\{Trop\}(f)=\backslash bigoplus\_\{i=1\}^s\; v(c\_i)\backslash otimes\; X^\{\backslash otimes\; A\_i\}$.

The set of points where a tropical polynomial $F$ is non-differentiable is called its associated tropical hypersurface, denoted $\backslash mathrm\{V\}(F)$ (in analogy to the vanishing set of a polynomial). Equivalently, $\backslash mathrm\{V\}(F)$ is the set of points where the minimum among the terms of $F$ is achieved at least twice. When $F=\backslash operatorname\{Trop\}(f)$ for a Laurent polynomial $f$, this latter characterization of $\backslash mathrm\{V\}(F)$ reflects the fact that at any solution to $f=0$, the minimum valuation of the terms of $f$ must be achieved at least twice in order for them all to cancel.{{cite book|last1=Maclagan|first1=Diane|author-link=Diane Maclagan|last2=Sturmfels|first2=Bernd |author2-link=Bernd Sturmfels|title= Introduction to Tropical Geometry |title-link= Introduction to Tropical Geometry |year=2015|publisher=American Mathematical Society|isbn=9780821851982 }}

For X an algebraic variety in the algebraic torus $(K^\{\backslash times\})^n$, the tropical variety of X or tropicalization of X, denoted $\backslash operatorname\{Trop\}(X)$, is a subset of $\backslash R^n$ that can be defined in several ways. The equivalence of these definitions is referred to as the Fundamental Theorem of Tropical Geometry.

Let $\backslash mathrm\{I\}(X)$ be the ideal of Laurent polynomials that vanish on X in $K[x\_1^\{\backslash pm\; 1\},\backslash ldots,x\_n^\{\backslash pm1\}]$. Define

:$\backslash operatorname\{Trop\}(X)=\backslash bigcap\_\{f\backslash in\backslash mathrm\{I\}(X)\}\backslash mathrm\{V\}(\backslash operatorname\{Trop\}(f))\backslash subseteq\backslash R^n$

When X is a hypersurface, its vanishing ideal $\backslash mathrm\{I\}(X)$ is a principal ideal generated by a Laurent polynomial f, and the tropical variety $\backslash operatorname\{Trop\}(X)$ is precisely the tropical hypersurface $\backslash mathrm\{V\}(\backslash operatorname\{Trop\}(f))$.

Every tropical variety is the intersection of a finite number of tropical hypersurfaces. A finite set of polynomials $\backslash \{f\_1,\backslash ldots,f\_r\backslash \}\backslash subseteq\backslash mathrm\{I\}(X)$ is called a tropical basis for X if $\backslash operatorname\{Trop\}(X)$ is the intersection of the tropical hypersurfaces of $\backslash operatorname\{Trop\}(f\_1),\backslash ldots,\backslash operatorname\{Trop\}(f\_r)$. In general, a generating set of $\backslash mathrm\{I\}(X)$ 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.

Choosing a vector $\backslash mathbf\{w\}$ in $\backslash R^n$ defines a map from the monomial terms of $K[x\_1^\{\backslash pm\; 1\},\backslash ldots,x\_n^\{\backslash pm1\}]$ to $\backslash R$ by sending the term m to $\backslash operatorname\{Trop\}(m)(\backslash mathbf\{w\})$. For a Laurent polynomial $f=m\_1+\backslash cdots+m\_s$, define the initial form of f to be the sum of the terms $m\_i$ of f for which $\backslash operatorname\{Trop\}(m\_i)(\backslash mathbf\{w\})$ is minimal. For the ideal $\backslash mathrm\{I\}(X)$, define its initial ideal with respect to $\backslash mathbf\{w\}$ to be

:$\backslash operatorname\{in\}\_\{\backslash mathbf\{w\}\}\backslash mathrm\{I\}(X)=(\backslash operatorname\{in\}\_\{\backslash mathbf\{w\}\}(f):f\backslash in\backslash mathrm\{I\}(X))$.

Then define

:$\backslash operatorname\{Trop\}(X)\; =\backslash \{\backslash mathbf\{w\}\backslash in\backslash R^n:\backslash operatorname\{in\}\_\{\backslash mathbf\{w\}\}\backslash mathrm\{I\}(X)\backslash neq(1)\backslash \}$.

Since we are working in the Laurent ring, this is the same as the set of weight vectors for which $\backslash operatorname\{in\}\_\{\backslash mathbf\{w\}\}\backslash mathrm\{I\}(X)$ does not contain a monomial.

When K has trivial valuation, $\backslash operatorname\{in\}\_\{\backslash mathbf\{w\}\}\backslash mathrm\{I\}(X)$ is precisely the initial ideal of $\backslash mathrm\{I\}(X)$ with respect to the monomial order given by a weight vector $\backslash mathbf\{w\}$. It follows that $\backslash operatorname\{Trop\}(X)$ is a subfan of the Gröbner fan of $\backslash mathrm\{I\}(X)$.

Suppose that X is a variety over a field K with valuation v whose image is dense in $\backslash R$ (for example a field of Puiseux series). By acting coordinate-wise, v defines a map from the algebraic torus $(K^\{\backslash times\})^n$ to $\backslash R^n$. Then define

:$\backslash operatorname\{Trop\}(X)=\backslash overline\{\backslash \{(v(x\_1),\backslash ldots,v(x\_n)):(x\_1,\backslash ldots,x\_n)\backslash in\; X\backslash \}\}$,

where the overline indicates the closure in the Euclidean topology. If the valuation of K is not dense in $\backslash R$, then the above definition can be adapted by extending scalars to larger field which does have a dense valuation.

This definition shows that $\backslash operatorname\{Trop\}(X)$ is the non-Archimedean amoeba over an algebraically closed non-Archimedean field K.{{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 | volume=3 | pages=257–300 | year=2004 | isbn=978-0-306-48657-9 | zbl=1072.14013 }}

If X is a variety over $\backslash Complex$, $\backslash operatorname\{Trop\}(X)$ can be considered as the limiting object of the amoeba $\backslash operatorname\{Log\}\_t(X)$ as the base t of the logarithm map goes to infinity.{{citation | first=Eric | last=Katz |author-link=Eric Katz| title=What is Tropical Geometry? | journal=Notices of the American Mathematical Society | volume=64 | issue=4 | pages=380–382 | url=https://www.ams.org/publications/journals/notices/201704/rnoti-p380.pdf |doi=10.1090/noti1507| year=2017 | doi-access=free }}

The following characterization describes tropical varieties intrinsically without reference to algebraic varieties and tropicalization. A set V in $\backslash R^n$ 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 $d-1$ 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 $d-1$.{{citation| first1=Dustin | last1=Cartwright | first2=Sam | last2=Payne | title=Connectivity of tropicalizations | journal=Mathematical Research Letters | volume=19 | issue=5 | year=2012 | pages=1089–1095 | doi=10.4310/MRL.2012.v19.n5.a10 | arxiv=1204.6589 | bibcode=2012arXiv1204.6589C | s2cid=51767353 }}

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.{{Cite journal|date=1 September 2013|title=Rank of divisors on tropical curves|journal=Journal of Combinatorial Theory, Series A| language=en|volume=120|issue=7|pages=1521–1538|doi=10.1016/j.jcta.2013.05.002|issn=0097-3165|last1=Hladký|first1=Jan|last2=Králʼ|first2=Daniel|last3=Norine|first3=Serguei|arxiv=0709.4485|s2cid=3045053}}

Many classical theorems of algebraic geometry have counterparts in tropical geometry, including:

• Pappus's hexagon theorem.{{Cite journal|last=Tabera|first=Luis Felipe|date=1 January 2005|title=Tropical constructive Pappus' theorem|journal=International Mathematics Research Notices| language=en|volume=2005|issue=39|pages=2373–2389|doi=10.1155/IMRN.2005.2373|issn=1073-7928|arxiv=math/0409126|doi-access=|s2cid=14250249 }}
• Bézout's theorem.
• The degree-genus formula.
• The Riemann–Roch theorem.{{Cite journal|last1=Kerber|first1=Michael|last2=Gathmann|first2=Andreas|date=1 May 2008|title=A Riemann–Roch theorem in tropical geometry|journal=Mathematische Zeitschrift| language=en|volume=259|issue=1|pages=217–230|doi=10.1007/s00209-007-0222-4|issn=1432-1823|arxiv=math/0612129|s2cid=15239772}}
• The group law of the cubics.{{cite book | last1=Chan | first1=Melody | author1-link = Melody Chan | last2=Sturmfels | first2=Bernd | author2-link=Bernd Sturmfels | 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 }}

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.

A tropical line appeared in Paul Klemperer's design of auctions used by the Bank of England during the financial crisis in 2007.{{Cite web|url = https://www.economics.ox.ac.uk/news/how-geometry-came-to-the-rescue-during-th |title = How geometry came to the rescue during the banking crisis|access-date = 24 March 2014|website = Department of Economics, University of Oxford}} 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 a subtropical convex algebra.{{Cite journal |doi = 10.1007/s40844-015-0012-3|title = International trade theory and exotic algebras |url=https://www.researchgate.net/publication/280646264 |journal = Evolutionary and Institutional Economics Review|volume = 12|pages = 177–212|year = 2015|last1 = Shiozawa|first1 = Yoshinori|issue = 1 |s2cid = 155827635 }} This is a digest of Y. Shiozawa, "[https://www.researchgate.net/publication/236020268 Subtropical Convex Geometry as the Ricardian Theory of International Trade]" draft paper.{{primary source inline|date=September 2024}} Tropical geometry has also been used for analyzing neural networks. One such result is that feedforward neural networks with ReLU activation are exactly tropical rational curves.{{cite conference|last1=Zhang|first1=Liwen|author-link1=|last2=Naitzat|first2=Gregory|author-link2=|last3=Lim|first3=Lek-Heng|author3-link=Lek-Heng Lim|date=2018|title=Tropical Geometry of Deep Neural Networks|url=http://proceedings.mlr.press/v80/zhang18i.html|conference=35th International Conference on Machine Learning|location=|publisher=|pages=5824–5832|id=|book-title=Proceedings of the 35th International Conference on Machine Learning}}

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.{{cite book |last= Krivulin |first= Nikolai |arxiv=1408.0313 |chapter= Tropical optimization problems |year=2014 |title=Advances in Economics and Optimization: Collected Scientific Studies Dedicated to the Memory of L. V. Kantorovich |pages=195–214 |publisher=Nova Science Publishers |location=New York |isbn=978-1-63117-073-7 |editor1=Leon A. Petrosyan |editor2=David W. K. Yeung |editor3=Joseph V. Romanovsky}} A tropical counterpart of the Abel–Jacobi map can be applied to a crystal design.{{cite book |author-link=Toshikazu Sunada|last=Sunada |first=T. |year=2012 |title=Topological Crystallography: With a View Towards Discrete Geometric Analysis |series=Surveys and Tutorials in the Applied Mathematical Sciences |volume=6 |publisher=Springer Japan |isbn=9784431541769}} The weights in a weighted finite-state transducer are often required to be a tropical semiring. Tropical geometry can show self-organized criticality.{{Cite journal|last1=Kalinin|first1=N.|last2=Guzmán-Sáenz|first2=A.|last3=Prieto|first3=Y.|last4=Shkolnikov|first4=M.|last5=Kalinina|first5=V.|last6=Lupercio|first6=E.|date=15 August 2018|title=Self-organized criticality and pattern emergence through the lens of tropical geometry|journal=Proceedings of the National Academy of Sciences of the United States of America| volume=115|issue=35|language=en|pages=E8135–E8142|doi=10.1073/pnas.1805847115|issn=0027-8424|pmid=30111541|pmc=6126730|arxiv=1806.09153|bibcode=2018PNAS..115E8135K|doi-access=free}}

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 {{cite journal| last = Tourkine| first = Piotr| title = Tropical Amplitudes| journal = Annales Henri Poincaré| volume = 18| issue = 6| pages = 2199–2249| date = 2017| doi = 10.1007/s00023-017-0560-7| arxiv = 1309.3551| bibcode = 2017AnHP...18.2199T}} and has found connections to constructions such as the Amplituhedron{{cite journal| last1 = Arkani-Hamed| first1 = Nima| last2 = Trnka| first2 = Jaroslav| title = The Amplituhedron| journal = JHEP| volume = 2014| pages = 030| date = 2014| issue = 10| doi = 10.1007/JHEP10(2014)030| arxiv = 1312.2007| bibcode = 2014JHEP...10..030A}} and tropological (topological Carrollian) sigma models.{{cite journal| last1 = Albrychiewicz| first1 = Emil| last2 = Ellers| first2 = Kai-Isaak| last3 = Franco Valiente| first3 = Andrés| last4 = Hořava| first4 = Petr| title = Tropological sigma models| journal = JHEP| volume = 06| pages = 135| date = 2024| issue = 6| doi = 10.1007/JHEP06(2024)135| arxiv = 2311.00745| bibcode = 2024JHEP...06..135A}}

• Tropical analysis
• Tropical compactification

{{reflist}}

{{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. 652–701).
• {{cite journal |last1=Bogart |first1=Tristram |last2=Jensen |first2=Anders |last3=Speyer |first3=David |last4=Sturmfels |first4=Bernd | author4-link=Bernd Sturmfels | last5=Thomas |first5=Rekha |arxiv=math/0507563 |title=Computing Tropical Varieties |year=2005 |doi=10.1016/j.jsc.2006.02.004 |volume=42 |issue=1–2 |journal=Journal of Symbolic Computation |pages=54–73|bibcode=2005math......7563B |s2cid=24788157 }}
• {{cite journal |last1=Einsiedler |first1=Manfred |last2=Kapranov |first2=Mikhail |last3=Lind |first3=Douglas |arxiv=math/0408311 |title=Non-archimedean amoebas and tropical varieties |journal=J. Reine Angew. Math. |volume=601 |pages=139–157 |year=2006 |bibcode=2004math......8311E }}
• {{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|last1=Maclagan|first1=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 | last1=Pachter| first1=Lior | author1-link = Lior Pachter | last2=Sturmfels | first2=Bernd | author2-link=Bernd Sturmfels | 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 | pmid=15534224 | pmc=528960 | zbl=1135.62302| arxiv=q-bio/0311009 | bibcode=2004PNAS..10116132P | doi-access=free }}
• {{cite arXiv |last= Speyer |first=David E. |eprint=math/0304218v3 |title=The Tropical Grassmannian |year=2003 }}
• {{cite journal |last1=Speyer |first1=David |last2=Sturmfels |first2=Bernd | author2-link=Bernd Sturmfels | arxiv=math/0408099 |title=Tropical Mathematics | orig-year=2004 | year= 2009 | zbl=1227.14051 | journal=Mathematics Magazine | volume=82 | number=3 | pages=163–173 | doi=10.4169/193009809x468760|s2cid=119142649 }}
• {{cite arXiv |last= Theobald |first= Thorsten |eprint=math/0306366v2 |title=First steps in tropical geometry |year=2003 }}

{{Refend}}

• {{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 | location=Providence, RI | publisher=American Mathematical Society | isbn=978-1-4704-1021-6 | year=2013 }}
• [https://web.archive.org/web/20220116040726/https://www.math.utah.edu/~yplee/teaching/7800f15/Gross_Kansas_cropped.pdf Tropical geometry and mirror symmetry]

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