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Introduction to Tropical Geometry

Diane Maclagan and Bernd Sturmfels
American Mathematical Society
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics 161
[Reviewed by
Felipe Zaldivar
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Born at the turn of the century, tropical algebraic geometry has developed quite rapidly as an area of research that is not just interesting by itself but also for its interactions with well-established areas of mathematics, from algebraic geometry and algebraic combinatorics on the pure side of mathematics to models in mathematical biology or dynamical programming.

The geometric objects of tropical geometry are roots of polynomials over the tropical semiring. As a set, the tropical semiring is the real field with an added infinity symbol (with the usual properties). Tropical addition is the minimum of the given (real) numbers and tropical multiplication is the sum of the given (real) numbers. The real number 0 is the tropical product identity and infinity is the tropical additive identity. Of course, tropical division is just the usual subtraction of (real) numbers and, in general, there is no tropical subtraction. One basic observation is that a tropical monomial is a linear function with integer coefficients and thus a tropical polynomial is a (finite, tropical) linear combination, with real coefficients, of these monomials. Hence, a tropical polynomial is a piecewise linear function and the roots of one such polynomial are, by definition, the points where it fails to be linear. Thus, a tropical variety looks like a family of convex objects in a Euclidean space.

Of course one must make all these intuitions precise to properly define a tropical variety, and the book under review does this in a systematic and engaging way. After a quick tour of some highlights of the subject in Chapter one, Chapter two begins by reviewing some basic facts from non-Arquimedean valued fields, classical algebraic varieties, polyhedral geometry, Gröbner bases over valued fields and a generalization to the ring of Laurent polynomials.

With these preliminaries, the main subjects of the book are introduced in Chapter three: the tropicalization of classical algebraic varieties over a valued field. We begin with its definition as intersection of the tropical hypersurfaces for all polynomials in an ideal of the ring of Laurent polynomials over the given field. The main goal is the Fundamental Theorem of tropical algebraic geometry, which provides equivalent definitions for a tropical variety. The connections between polyhedral and tropical geometry are also explored in this chapter, including as the main results the Structure Theorem on which polyhedral complexes are tropical varieties, and the connectivity properties of the polyhedral complexes supported on a tropical variety. The last section of this chapter is devoted to the important question of intersection of tropicalizations of algebraic varieties.

The remaining chapters explore families of tropical varieties and connections to other classical subjects such as enumerative geometry or degenerations. Thus, Chapter four looks at the tropical analogs of linear spaces and their parameter spaces, the tropical Grassmannians, for example. Matroids, which are important in this context, are also introduced in this chapter. Chapter five further explores the connections with linear algebra in its tropical guise, from eigenvalues and eigenvectors to tropical convexity and determinantal varieties. The last chapter explores the deep connections between tropical geometry and the theory of toric varieties of classical algebraic geometry, deeply tied from their very definition to polyhedral geometry.

As the sketch above tries to convey, tropical geometry is a rich subject with deep ties to several areas of classical mathematics. The book under review makes the subject accessible and enjoyable, requiring only a minimal background on algebra. The book develops the theory in a self-contained way, adding plenty of examples to illustrate or highlight some points, with detailed computations and wonderful figures. Each chapter comes with set of problems to test the potential reader grasp of the subject. There are not many introductory books on this subject, perhaps only the Oberwolfach Seminar Tropical Algebraic Geometry by Itenberg, Mikhalkin and Shustin (Second Edition, Birkhäuser, 2009) address to advanced graduate students. The book under review is a beautiful addition to the successful Graduate Studies in Mathematics textbooks of the AMS.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is