This book is about the interplay of computational commutative algebra and the theory of convex polytopes. It centers around a special class of ideals in a polynomial ring: the class of toric ideals. They are characterized as those prime ideals that are generated by monomial differences or as the defining ideals of toric varieties (not necessarily normal).

The interdisciplinary nature of the study of Gröbner bases is reflected by the specific applications appearing in this book. These applications lie in the domains of integer programming and computational statistics. The mathematical tools presented in the volume are drawn from commutative algebra, combinatorics, and polyhedral geometry.

Bernd Sturmfels, University of California, Berkeley, Berkeley, CA

Chapter 1. Gröbner basics
Chapter 2. The state polytope
Chapter 3. Variation of term orders
Chapter 4. Toric ideals
Chapter 5. Enumeration, sampling and integer programming
Chapter 6. Primitive partition identities
Chapter 7. Universal Gröbner bases
Chapter 8. Regular triangulations
Chapter 9. The second hypersimplex
Chapter 10. 𝒜-graded algebras
Chapter 11. Canonical subalgebra bases
Chapter 12. Generators, Betti numbers and localizations
Chapter 13. Toric varieties in algebraic geometry
Chapter 14. Some specific Gröbner bases