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.
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Editorial Reviews
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"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
"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 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 ---- Mathematical Reviews
"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 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 ---- Mathematical Reviews
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3 of 3 people found the following review helpful:
5.0 out of 5 stars
A dense, but excellent, text,
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This review is from: Grobner Bases and Convex Polytopes (University Lecture Series, No. 8) (Paperback)
I cannot recommend this text highly enough to anyone who is working with Grobner bases, especially in computational commutative algebra. Toric ideals, the state polytope, several algorithms for computing Grobner bases -- Sturmfels brings all this together, as well as showing how it can all be applied to problems such as that of integer programming.Thanks to his excellent taste in the theorems he presents, Sturmfels has managed to put all this together in a relatively small, well-written package. However, his brevity has made this a very dense text -- you would do well to keep a copy of Schrijver's Theory of Linear and Integer Programming at hand for proofs of some of the theorems, and if you are like me (a first year grad student) you will invariably need to look in some other algebraic and geometric texts for a more thorough treatment of some of the topics in here.
1 of 10 people found the following review helpful:
1.0 out of 5 stars
why bother?,
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This review is from: Grobner Bases and Convex Polytopes (University Lecture Series, No. 8) (Paperback)
Another impenetrable text from this author.
Too bad, it was so unnecessary as the subject easily lends itself to user friendly exposition.
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Inside This Book
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First Sentence:
Let k be any field and k[x] = k[x1, . . . . , xn] the polynomial ring in n indeterminates. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
primitive partition identities, state polytope, primitive partition identity, distinct initial ideals, lexicographic triangulation, second hypersimplex, toric ideals, lexicographic term order, primitive binomial, monomial ideal, secondary polytope, binomial ideal, polytope conv, nilpotent modulo, polyhedral subdivision, projective toric variety, grading deg, minimal generator, toric surface, affine toric variety, standard monomials, normal fan, regular triangulation, affine span, twisted cubic curve New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Let k be any field and k[x] = k[x1, . . . . , xn] the polynomial ring in n indeterminates. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
primitive partition identities, state polytope, primitive partition identity, distinct initial ideals, lexicographic triangulation, second hypersimplex, toric ideals, lexicographic term order, primitive binomial, monomial ideal, secondary polytope, binomial ideal, polytope conv, nilpotent modulo, polyhedral subdivision, projective toric variety, grading deg, minimal generator, toric surface, affine toric variety, standard monomials, normal fan, regular triangulation, affine span, twisted cubic curve New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 14
books:
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1
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cites this book:
- Algebraic Geometry (Graduate Texts in Mathematics) by Robin Hartshorne on page 135, and Back Matter
- Algorithms in Combinatorial Geometry (Eatcs Monographs on Theoretical Computer Science) by H. Edelsbrunner in Back Matter
- Convex Bodies and Algebraic Geometry: An Introduction to the Theory of Toric Varieties (Ergebnisse Der Mathematik Und Ihrer Grenzgebiete 3 Folge) by Tadao Oda in Back Matter
- Algorithms for Random Generation and Counting: A Markov Chain Approach (Progress in Theoretical Computer Science) by A. Sinclair in Back Matter
- Algorithms in Invariant Theory (Archives of Virology) by Bernd Sturmfels in Back Matter
See all 14 books this book cites
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