Softcover ISBN:  9780821804872 
Product Code:  ULECT/8 
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AMS Member Price:  $32.80 
Electronic ISBN:  9781470421571 
Product Code:  ULECT/8.E 
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Book DetailsUniversity Lecture SeriesVolume: 8; 1996; 162 ppMSC: Primary 13; 14; Secondary 52; 90;
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.ReadershipGraduate students and mathematicians interested in computer science and theoretical operations research.

Table of Contents

Chapters

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. $\mathcal {A}$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


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 
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


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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.
Graduate students and mathematicians interested in computer science and theoretical operations research.

Chapters

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. $\mathcal {A}$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

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 
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