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. Readership Graduate students and mathematicians interested in computer science and theoretical operations research. Reviews "This book is a stateoftheart account of the rich interplay between combinatorics and geometry of convex polytopes and computational commutative algebra via the tool of Gröbner bases. It is an essential introduction for those who wish to perform research in this fastdeveloping, interdisciplinary field. For the math programmer, this book could be viewed as an exposition of the interactions between integer programming and Gröbner bases."  Optima "Material is presented in a concise way ... lots of motivating examples ... not only of interest for mathematicians studying Gröbner bases, but also for researchers working on the mathematical aspects of integer programming and computational statistics."  Newsletter on Computational and Applied Mathematics "Thanks to the author's ingenious writing, most of the material should be accessible to firstyear graduate students in mathematics ... will be a landmark for further study of Gröbner bases in new branches of mathematics. It underlines the powerful techniques of commutative algebra in the interplay with combinatorics and polyhedral geometry."  Mathematical Reviews "The methods discussed in the book lead to substantial conceptual insights."  Zentralblatt MATH "The exposition is clear and very well motivated. There is an abundance of illustrative examples; often, the same example is carried through a number of chapters to give coherence to the discussion ... The reader will be amply rewarded, as this is an elegantly written work of wide scholarship."  Bulletin of the London Mathematical Society "This monograph represents a well written introduction to a rapidly developing field of algebra. The exercises and bibliographical remarks included will make it easy for the reader keen on understanding the interplay between commutative algebra and the subjects quoted above to gain deeper insight."  Monatshefte für Mathematik Table of Contents  Gröbner basics
 The state polytope
 Variation of term orders
 Toric ideals
 Enumeration, sampling and integer programming
 Primitive partition identities
 Universal Gröbner bases
 Regular triangulations
 The second hypersimplex
 \(\mathcal A\)graded algebras
 Canonical subalgebra bases
 Generators, Betti numbers and localizations
 Toric varieties in algebraic geometry
 Some specific Gröbner bases
 Bibliography
 Index
