# Tropical geometry

Tropical geometry is a relatively new area in mathematics, which might loosely be described as a piece-wise linear or skeletonized version of algebraic geometry, using the tropical semiring instead of a field.

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 classic results from algebraic geometry, such as the Brill–Noether theorem, using the tools of tropical geometry.

## History

The basic ideas of tropical analysis have been developed independently in the same notations by mathematicians working in various fields. The leading ideas of tropical geometry had appeared in different forms in the 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 has been motivated by the applications to enumerative algebraic geometry, with ideas from Maxim Kontsevich and works by Grigory Mikhalkin among others.

The adjective tropical in the name of the area 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, 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 (ℝ ∪ {+∞}, ⊕, ⊗), with the operations:

$x\oplus y=\min\{x,y\},$ $x\otimes y=x+y.$ The operations ⊕ and ⊗ are referred to as tropical addition and tropical multiplication respectively. The unit for ⊕ is +∞, and the unit for ⊗ is 0.

Similarly, the max tropical semiring is the semiring (ℝ ∪ {−∞}, ⊕, ⊗), with operations:

$x\oplus y=\max\{x,y\},$ $x\otimes y=x+y.$ The unit for ⊕ is −∞, and the unit for ⊗ is 0.

These semirings are isomorphic, under negation $x\mapsto -x$ , 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 valuated fields encountered in tropical geometry are:

• Q or C with the trivial valuation, v(a) = 0 for all a ≠ 0,
• Q or its extensions with the p-adic valuation, v(pna/b) = n for a and b coprime to p,
• the field of Laurent series C((t)) (integer powers), or the field of (complex) Puiseux series C{{t}}, with valuation returning the smallest exponent of t appearing in the series.

## Tropical polynomials

A tropical polynomial is a function F: RnR 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 X1,...,Xn. 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.

{\begin{aligned}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{aligned}} Given a polynomial f in Laurent polynomial ring K[x1±,...,xn±] where K is a valued field, the tropicalization of 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, denoted Trop(f).

$f=\sum _{i=1}^{s}c_{i}x^{A_{i}}\quad {\text{ with }}A_{1},\ldots ,A_{s}\in \mathbb {Z} ^{n},$ $\operatorname {Trop} (f)=\bigoplus _{i=1}^{s}v(c_{i})\otimes X^{\otimes A_{i}}.$ The set of points where a tropical polynomial F is non-differentiable is called its associated tropical hypersurface, denoted V(F) (in analogy to the vanishing set of a polynomial). Equivalently, V(F) is the set of points where the minimum among the terms of F is achieved at least twice. When F = Trop(f) for a Laurent polynomial f, this latter characterization of 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.

## Tropical varieties

### Definitions

For X an algebraic variety in the algebraic torus (K×)n, the tropical variety of X or tropicalization of X, denoted Trop(X), is a subset of Rn that can be defined in several ways. The equivalence of these definitions is referred to as the Fundamental Theorem of Tropical Geometry.

#### Intersection of tropical hypersurfaces

Let I(X) be the ideal of Laurent polynomials that vanish on X in K[x1±,...,xn±]. Define

$\operatorname {Trop} (X)=\bigcap _{f\in \mathrm {I} (X)}\mathrm {V} (\operatorname {Trop} (f))\subseteq \mathbb {R} ^{n}.$ When X is a hypersurface, its vanishing ideal I(X) is a principal ideal generated by a Laurent polynomial f, and the tropical variety Trop(X) is precisely the tropical hypersurface V(Trop(f)).

Every tropical variety is the intersection of a finite number of tropical hypersurfaces. A finite set of polynomials $\{f_{1},\ldots ,f_{r}\}\subseteq \mathrm {I} (X)$ is called a tropical basis for X if Trop(X) is the intersection of the tropical hypersurfaces of $\operatorname {Trop} (f_{1}),\ldots ,\operatorname {Trop} (f_{r})$ . In general, a generating set of 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.

#### Initial ideals

Choosing a vector w in Rn defines a map from the monomial terms of K[x1±,...,xn±] to R by sending term m to Trop(m)(w). For a Laurent polynomial f = m1 + ... + ms, define the initial form of f to be the sum of the terms mi of f for which Trop(mi)(w) is minimal. For ideal I(X), define its initial ideal with respect to w to be

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

$\operatorname {Trop} (X)=\{\mathbf {w} \in \mathbb {R} ^{n}:\operatorname {in} _{\mathbf {w} }\mathrm {I} (X)\neq (1)\}.$ Since we are working in the Laurent ring, this is equivalent to the set weight vectors for which inwI(X) does not contain a monomial.

When K has trivial valuation, inwI(X) is precisely the initial ideal of I(X) with respect to the monomial order given by weight vector w. It follows that Trop(X) is a subfan of the Gröbner fan of I(X).

#### Image of the valuation map

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

$\operatorname {Trop} (X)={\overline {\{(v(x_{1}),\ldots ,v(x_{n})):(x_{1},\ldots ,x_{n})\in X\}}},$ where the overline indicates the closure in the Euclidean topology. If the valuation of K is not dense in R, then the above definition can be adapted by extending scalars to larger field which does have a dense valuation.

This definition shows that Trop(X) is the non-Archimedean amoeba over an algebraically closed non-Archimedean field K.

If X is a variety over C, Trop(X) can be considered as the limiting object of the amoeba $\operatorname {Log} _{t}(X)$ as the base t of the logarithm map goes to infinity.

#### Polyhedral complex

The following characterization describes tropical varieties intrinsically without reference to algebraic varieties and tropicalization. A set V in Rn 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.

### Tropical curves

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.

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

Oleg Viro used tropical curves to classify real curves in the plane up to isotopy of degree 7. His method of patchworking gives a procedure to build a real curve of a given isotopy class from its tropical curve.

## Applications

A tropical line appeared in Paul Klemperer's design of auctions used by the Bank of England during the financial crisis in 2007. 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 subtropical convex algebra.

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. A tropical counterpart of the Abel–Jacobi map can be applied to a crystal design. The weights in a weighted finite-state transducer are often required to be a tropical semiring. Tropical geometry shows self-organized criticality behaviour.