**University Lecture Series**

Volume: 8;
1996;
162 pp;
Softcover

MSC: Primary 13; 14;
Secondary 52; 90

Print ISBN: 978-0-8218-0487-2

Product Code: ULECT/8

List Price: $38.00

Individual Member Price: $30.40

**Electronic ISBN: 978-1-4704-2157-1
Product Code: ULECT/8.E**

List Price: $38.00

Individual Member Price: $30.40

# Gröbner Bases and Convex Polytopes

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*Bernd Sturmfels*

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.

#### Table of Contents

# Table of Contents

## Grobner Bases and Convex Polytopes

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Introduction ix10 free
- Chapter 1. Gröbner Basics 114 free
- Chapter 2. The State Poly tope 922
- Chapter 3. Variation of Term Orders 1932
- Chapter 4. Toric Ideals 3144
- Chapter 5. Enumeration, Sampling and Integer Programming 3952
- Chapter 6. Primitive Partition Identities 4760
- Chapter 7. Universal Grobner Bases 5568
- Chapter 8. Regular Triangulations 6376
- Chapter 9. The Second Hypersimplex 7588
- Chapter 10. A-graded Algebras 8598
- Chapter 11. Canonical Subalgebra Bases 99112
- Chapter 12. Generators, Betti Numbers and Localizations 113126
- Chapter 13. Toric Varieties in Algebraic Geometry 127140
- Chapter 14. Some Specific Grobner Bases 141154
- Bibliography 155168
- Index 161174
- Back Cover Back Cover1176

#### Readership

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

#### Reviews

This book is a state-of-the-art 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 fast-developing, 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 first-year 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