Tropical geometry

A tropical cubic curve

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. Its leading ideas had appeared in different guises in previous works of George M. Bergman and of Robert Bieri and John Groves, but only since the late 1990's has an effort been made to consolidate the basic definitions of the theory. This effort has been in great part motivated by the strong applications to enumerative algebraic geometry uncovered by Grigory Mikhalkin.

The adjective tropical was supposedly coined by French mathematicians in honor of the Hungarian-born Brazilian mathematician Imre Simon, who pioneered the field. (The word “tropical” being the French view of Brazil?) However, it is surprisingly difficult to verify this explanation. Speyer and Sturmfels[1] attribute it to Jean-Eric Pin,[2] who again blames Dominique Perrin, while Simon[3] himself attributes the word to yet a third French mathematician, Christian Choffrut.

Basic definitions

We will 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.

Consider the tropical semiring (also known as the min-plus algebra due to the definition of the semiring). This semiring, (ℝ ∪ {∞}, ⊕, ⊗), is defined with the operations as follows:

$x \oplus y = \min\{\, x, y \,\},\,$
$x \otimes y = x + y.\,$

From this we can also define tropical exponentiation in the usual way as iterated tropical products.

A monomial of variables in this semiring is a linear map, represented in classical arithmetic as a linear function of the variables with integer coefficients.[1] A polynomial in the semiring is the minimum of a finite number of such monomials, and is therefore a concave, continuous, 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:

1. Tropical hypersurfaces are exactly the rational polyhedral complexes satisfying a "zero-tension" condition.[1]
2. Tropical surfaces are exactly the non-Archimedean amoebas over an algebraically closed non-archimedean field K.[4]

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 K[x1, ..., xn] instead of polynomials. It has been proved that 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. The study of tropical curves (tropical hypersurfaces in ℝ2) is particularly well developed. In fact, for this setting, mathematicians have established analogues of many classical theorems; e.g., Pappus's theorem[disambiguation needed], Bézout's theorem, the degree-genus formula, and the group law of the cubics[5] all have tropical counterparts.

Applications

Tropical geometry was used by Economist Paul Klemperer to design auctions used by the Bank of England during the financial crisis in 2007.[6] 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.[7]

References

1. ^ a b c David Speyer and Bernd Sturmfels. Tropical mathematics. Mathematics Magazine 82.3 (2009), pp. 163–173.
2. ^ Jean-Eric Pin. Tropical semirings. Idempotency (Bristol, 1994). Publ. Newton Inst 11 (1998), pp. 50–69.
3. ^ Imre Simon. Recognizable sets with multiplicities in the tropical semiring. Mathematical Foundations of Computer Science (1988), pp. 107–120.
4. ^ Mikhalkin, Grigory (2004). "Amoebas of algebraic varieties and tropical geometry". In Donaldson, Simon; Eliashberg, Yakov; Gromov, Mikhael. Different faces of geometry. International Mathematical Series (New York) 3. New York, NY: Kluwer Academic/Plenum Publishers. pp. 257–300. ISBN 0-306-48657-1. Zbl 1072.14013.
5. ^ Chan, Melody; Sturmfels, Bernd (2013). "Elliptic curves in honeycomb form". In Brugallé, Erwan. 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. Contemporary Mathematics 589. Providence, RI: American Mathematical Society. pp. 87–107. arXiv:1203.2356. ISBN 978-0-8218-9146-9. Zbl 06241528.
6. ^ "How geometry came to the rescue during the banking crisis". Department of Economics, University of Oxford. Retrieved 24 March 2014.
7. ^ Y. Shiozawa, "Subtropical Convex Geometry as the Ricardian Theory of International Trade," draft paper in his ResearchGate page