##
**Introduction to tropical geometry.**
*(English)*
Zbl 1321.14048

Graduate Studies in Mathematics 161. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-5198-2/hbk). xii, 363 p. (2015).

From the Preface:

“Tropical geometry is an exciting new field at the interface between algebraic geometry and combinatorics with connections to many other areas. At its heart it is geometry over the tropical semiring, which is \(\mathbb R\cup\{\infty\}\) with the usual operations of addition and multiplication replaced by minimum and addition, respectively. This turns polynomials into piecewise-linear functions and replaces an algebraic variety by an object from polyhedral geometry, which can be regarded as a ‘combinatorial shadow’ of the original variety.

In this book we introduce this subject at a level that is accessible to beginners. Tropical geometry has become a large field, and only a small selection of topics can be covered in a first course. We focus on the study of tropical varieties that arise from classical algebraic varieties. Methods from commutative algebra and polyhedral geometry are central to our approach. This necessarily means that many important topics are left out. These include the systematic development of tropical geometry as an intrinsic geometry in its own right, connections to enumerative and real algebraic geometry, connections to mirror symmetry, connections to Berkovich spaces and abstract curves, and the more applied aspects of max-plus algebra. Luckily most of these topics are covered in other recent or forthcoming books, such as [F. L. Baccelli et al., Synchronization and linearity. An algebra for discrete event systems. Chichester: Wiley (1992; Zbl 0824.93003)], [P. Butkovič, Max-linear systems. Theory and algorithms. London: Springer (2010; Zbl 1202.15032)], [M. Gross, Tropical geometry and mirror symmetry. Providence, RI: American Mathematical Society (AMS) (2011; Zbl 1215.14061)], [I. Itenberg et al., Tropical algebraic geometry. Basel: Birkhäuser (2007; Zbl 1162.14300)], [M. Joswig, Essentials of tropical combinatorics. Book in preparation], [G. Mikhalkin and J. Rau, Tropical geometry, Book in preparation], and [L. Pachter (ed.) and B. Sturmfels (ed.), Algebraic statistics for computational biology. Cambridge: Cambridge University Press (2005; Zbl 1108.62118)].

Prerequisites. This text is intended to be suitable for a class on tropical geometry for first-year graduate students in mathematics. We have attempted to make the material accessible to readers with a minimal background in algebraic geometry, at the level of the undergraduate text book Ideals, Varieties, and Algorithms by D. Cox et al. [Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. 3rd ed. New York, NY: Springer (2007; Zbl 1118.13001)].

Essential prerequisites for this book are mastery of linear algebra and the material on rings and fields in a first course in abstract algebra. Since tropical geometry draws on many fields of mathematics, some additional background in geometry, topology, or number theory will be beneficial.

Polyhedra and polytopes play a fundamental role in tropical geometry, and some prior exposure to convexity and polyhedral combinatorics may help. For that we recommend Ziegler’s book Lectures on Polytopes [G. M. Ziegler, Lectures on polytopes. Berlin: Springer-Verlag (1995; Zbl 0823.52002)].

Chapter 1 offers a friendly welcome to our readers. It has no specific prerequisites and is meant to be enjoyable for all. The first three sections in Chapter 2 cover background material in abstract algebra, algebraic geometry, and polyhedral geometry. Enough definitions and examples are given that an enthusiastic reader can fill in any gaps. All students (and their teachers) are strongly urged to explore the exercises for Chapters 1 and 2.

Some of the results and their proofs will demand more mathematical maturity and expertise. Chapters 2 and 3 require some commutative algebra. Combinatorics and multilinear algebra will be useful for studying Chapters 4 and 5. Chapter 6 assumes familiarity with modern algebraic geometry.

Overview. We begin by relearning the arithmetic operations of addition and multiplication. The rest of Chapter 1 offers tapas that can be enjoyed in any order. They show a glimpse of the past, present, and future of tropical geometry and serve as an introduction to the more formal contents of this book. In Chapter 2, the first half covers background material, while the second half develops a version of Gröbner basis theory suitable for algebraic varieties over a field with valuation. The highlights are the construction of the Gröbner complex and the resulting finiteness of tropical bases.

Chapter 3 is the heart of the book. The two main results are the Fundamental Theorem 3.2.3, which characterizes tropical varieties in seemingly different ways, and the Structure Theorem 3.3.5, which says that they are connected balanced polyhedral complexes of the correct dimension. Stable intersections of tropical varieties reveal a hint of intersection theory.

Tropical linear spaces and their parameter spaces, the Grassmannian and the Dressian, appear in Chapter 4. Matroid theory plays a foundational role. Our discussion of complete intersections includes mixed volumes of Newton polytopes and a tropical proof of Bernstein’s Theorem for n equations in n variables. We also study the combinatorics of surfaces in 3-space.

Chapter 5 covers spectral theory for tropical matrices, tropical convexity, and determinantal varieties. It also showcases computations with Bergman fans of matroids and other linear spaces. Chapter 6 concerns the connection between tropical varieties and toric varieties. It introduces the tropical approach to degenerations, compactifications, and enumerative geometry.”

“Tropical geometry is an exciting new field at the interface between algebraic geometry and combinatorics with connections to many other areas. At its heart it is geometry over the tropical semiring, which is \(\mathbb R\cup\{\infty\}\) with the usual operations of addition and multiplication replaced by minimum and addition, respectively. This turns polynomials into piecewise-linear functions and replaces an algebraic variety by an object from polyhedral geometry, which can be regarded as a ‘combinatorial shadow’ of the original variety.

In this book we introduce this subject at a level that is accessible to beginners. Tropical geometry has become a large field, and only a small selection of topics can be covered in a first course. We focus on the study of tropical varieties that arise from classical algebraic varieties. Methods from commutative algebra and polyhedral geometry are central to our approach. This necessarily means that many important topics are left out. These include the systematic development of tropical geometry as an intrinsic geometry in its own right, connections to enumerative and real algebraic geometry, connections to mirror symmetry, connections to Berkovich spaces and abstract curves, and the more applied aspects of max-plus algebra. Luckily most of these topics are covered in other recent or forthcoming books, such as [F. L. Baccelli et al., Synchronization and linearity. An algebra for discrete event systems. Chichester: Wiley (1992; Zbl 0824.93003)], [P. Butkovič, Max-linear systems. Theory and algorithms. London: Springer (2010; Zbl 1202.15032)], [M. Gross, Tropical geometry and mirror symmetry. Providence, RI: American Mathematical Society (AMS) (2011; Zbl 1215.14061)], [I. Itenberg et al., Tropical algebraic geometry. Basel: Birkhäuser (2007; Zbl 1162.14300)], [M. Joswig, Essentials of tropical combinatorics. Book in preparation], [G. Mikhalkin and J. Rau, Tropical geometry, Book in preparation], and [L. Pachter (ed.) and B. Sturmfels (ed.), Algebraic statistics for computational biology. Cambridge: Cambridge University Press (2005; Zbl 1108.62118)].

Prerequisites. This text is intended to be suitable for a class on tropical geometry for first-year graduate students in mathematics. We have attempted to make the material accessible to readers with a minimal background in algebraic geometry, at the level of the undergraduate text book Ideals, Varieties, and Algorithms by D. Cox et al. [Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. 3rd ed. New York, NY: Springer (2007; Zbl 1118.13001)].

Essential prerequisites for this book are mastery of linear algebra and the material on rings and fields in a first course in abstract algebra. Since tropical geometry draws on many fields of mathematics, some additional background in geometry, topology, or number theory will be beneficial.

Polyhedra and polytopes play a fundamental role in tropical geometry, and some prior exposure to convexity and polyhedral combinatorics may help. For that we recommend Ziegler’s book Lectures on Polytopes [G. M. Ziegler, Lectures on polytopes. Berlin: Springer-Verlag (1995; Zbl 0823.52002)].

Chapter 1 offers a friendly welcome to our readers. It has no specific prerequisites and is meant to be enjoyable for all. The first three sections in Chapter 2 cover background material in abstract algebra, algebraic geometry, and polyhedral geometry. Enough definitions and examples are given that an enthusiastic reader can fill in any gaps. All students (and their teachers) are strongly urged to explore the exercises for Chapters 1 and 2.

Some of the results and their proofs will demand more mathematical maturity and expertise. Chapters 2 and 3 require some commutative algebra. Combinatorics and multilinear algebra will be useful for studying Chapters 4 and 5. Chapter 6 assumes familiarity with modern algebraic geometry.

Overview. We begin by relearning the arithmetic operations of addition and multiplication. The rest of Chapter 1 offers tapas that can be enjoyed in any order. They show a glimpse of the past, present, and future of tropical geometry and serve as an introduction to the more formal contents of this book. In Chapter 2, the first half covers background material, while the second half develops a version of Gröbner basis theory suitable for algebraic varieties over a field with valuation. The highlights are the construction of the Gröbner complex and the resulting finiteness of tropical bases.

Chapter 3 is the heart of the book. The two main results are the Fundamental Theorem 3.2.3, which characterizes tropical varieties in seemingly different ways, and the Structure Theorem 3.3.5, which says that they are connected balanced polyhedral complexes of the correct dimension. Stable intersections of tropical varieties reveal a hint of intersection theory.

Tropical linear spaces and their parameter spaces, the Grassmannian and the Dressian, appear in Chapter 4. Matroid theory plays a foundational role. Our discussion of complete intersections includes mixed volumes of Newton polytopes and a tropical proof of Bernstein’s Theorem for n equations in n variables. We also study the combinatorics of surfaces in 3-space.

Chapter 5 covers spectral theory for tropical matrices, tropical convexity, and determinantal varieties. It also showcases computations with Bergman fans of matroids and other linear spaces. Chapter 6 concerns the connection between tropical varieties and toric varieties. It introduces the tropical approach to degenerations, compactifications, and enumerative geometry.”

Reviewer: Péter E. Frenkel (Budapest)

### MSC:

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |

14T05 | Tropical geometry (MSC2010) |

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |

13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |

52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |