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Grobner Bases and Convex Polytopes

Grobner Bases and Convex Polytopes PDF Author: Bernd Sturmfels
Publisher: American Mathematical Soc.
ISBN: 0821804871
Category : Mathematics
Languages : en
Pages : 176

Book Description
This book is about the interplay of computational commutative algebra and the theory of convex polytopes. It centres 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.

Grobner Bases and Convex Polytopes

Grobner Bases and Convex Polytopes PDF Author: Bernd Sturmfels
Publisher: American Mathematical Soc.
ISBN: 0821804871
Category : Mathematics
Languages : en
Pages : 176

Book Description
This book is about the interplay of computational commutative algebra and the theory of convex polytopes. It centres 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.

Gröbner Bases and Convex Polytopes

Gröbner Bases and Convex Polytopes PDF Author: Bernd Sturmfels
Publisher: American Mathematical Soc.
ISBN: 9781470421571
Category : Mathematics
Languages : en
Pages : 162

Book Description
This work is about the interplay of computational commutative algebra and the theory of convex polytopes. It centres 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 Grobner 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.

Gröbner Bases

Gröbner Bases PDF Author: Takayuki Hibi
Publisher: Springer Science & Business Media
ISBN: 4431545743
Category : Mathematics
Languages : en
Pages : 488

Book Description
The idea of the Gröbner basis first appeared in a 1927 paper by F. S. Macaulay, who succeeded in creating a combinatorial characterization of the Hilbert functions of homogeneous ideals of the polynomial ring. Later, the modern definition of the Gröbner basis was independently introduced by Heisuke Hironaka in 1964 and Bruno Buchberger in 1965. However, after the discovery of the notion of the Gröbner basis by Hironaka and Buchberger, it was not actively pursued for 20 years. A breakthrough was made in the mid-1980s by David Bayer and Michael Stillman, who created the Macaulay computer algebra system with the help of the Gröbner basis. Since then, rapid development on the Gröbner basis has been achieved by many researchers, including Bernd Sturmfels. This book serves as a standard bible of the Gröbner basis, for which the harmony of theory, application, and computation are indispensable. It provides all the fundamentals for graduate students to learn the ABC’s of the Gröbner basis, requiring no special knowledge to understand those basic points. Starting from the introductory performance of the Gröbner basis (Chapter 1), a trip around mathematical software follows (Chapter 2). Then comes a deep discussion of how to compute the Gröbner basis (Chapter 3). These three chapters may be regarded as the first act of a mathematical play. The second act opens with topics on algebraic statistics (Chapter 4), a fascinating research area where the Gröbner basis of a toric ideal is a fundamental tool of the Markov chain Monte Carlo method. Moreover, the Gröbner basis of a toric ideal has had a great influence on the study of convex polytopes (Chapter 5). In addition, the Gröbner basis of the ring of differential operators gives effective algorithms on holonomic functions (Chapter 6). The third act (Chapter 7) is a collection of concrete examples and problems for Chapters 4, 5 and 6 emphasizing computation by using various software systems.

Minkowski Addition of Polytopes

Minkowski Addition of Polytopes PDF Author: Peter Gritzmann
Publisher:
ISBN:
Category :
Languages : en
Pages : 62

Book Description


Harmony of Gröbner Bases and the Modern Industrial Society

Harmony of Gröbner Bases and the Modern Industrial Society PDF Author: Takayuki Hibi
Publisher: World Scientific
ISBN: 9814383457
Category : Mathematics
Languages : en
Pages : 385

Book Description
This volume consists of research papers and expository survey articles presented by the invited speakers of the conference on "Harmony of Grobner Bases and the Modern Industrial Society." Topics include computational commutative algebra, algebraic statistics, algorithms of D-modules and combinatorics. This volume also provides current trends on Grobner bases and will stimulate further development of many research areas surrounding Gr bner bases. Contents: Polyhedral Approach to Statistical Learning Graphical Models; Implementation of a Primary Decomposition Package; Computing Tropical Resultants; Running Markov Chain Without Markov Basis; Incomplete A-Hypergeometric Systems; Degree Bounds for a Minimal Markov Basis for the Three-State Toric Homogeneous Markov Chain Model.

Harmony Of Grobner Bases And The Modern Industrial Society - The Second Crest-sbm International Conference

Harmony Of Grobner Bases And The Modern Industrial Society - The Second Crest-sbm International Conference PDF Author: Takayuki Hibi
Publisher: World Scientific
ISBN: 9814452947
Category : Mathematics
Languages : en
Pages : 385

Book Description
This volume consists of research papers and expository survey articles presented by the invited speakers of the conference on “Harmony of Gröbner Bases and the Modern Industrial Society”. Topics include computational commutative algebra, algebraic statistics, algorithms of D-modules and combinatorics. This volume also provides current trends on Gröbner bases and will stimulate further development of many research areas surrounding Gröbner bases./a

Convex Polytopes

Convex Polytopes PDF Author: P. McMullen
Publisher: CUP Archive
ISBN: 9780521080170
Category : Mathematics
Languages : en
Pages : 196

Book Description


Harmony of Gr”bner Bases and the Modern Industrial Society

Harmony of Gr”bner Bases and the Modern Industrial Society PDF Author: Takayuki Hibi
Publisher: World Scientific
ISBN: 9814383465
Category : Mathematics
Languages : en
Pages : 385

Book Description
This volume consists of research papers and expository survey articles presented by the invited speakers of the conference on OC Harmony of GrAbner Bases and the Modern Industrial SocietyOCO. Topics include computational commutative algebra, algebraic statistics, algorithms of D-modules and combinatorics. This volume also provides current trends on GrAbner bases and will stimulate further development of many research areas surrounding GrAbner bases."

Gröbner Bases and Applications

Gröbner Bases and Applications PDF Author: Bruno Buchberger
Publisher: Cambridge University Press
ISBN: 9780521632980
Category : Mathematics
Languages : en
Pages : 566

Book Description
Comprehensive account of theory and applications of Gröbner bases, co-edited by the subject's inventor.

Binomial Ideals

Binomial Ideals PDF Author: Jürgen Herzog
Publisher: Springer
ISBN: 3319953494
Category : Mathematics
Languages : en
Pages : 332

Book Description
This textbook provides an introduction to the combinatorial and statistical aspects of commutative algebra with an emphasis on binomial ideals. In addition to thorough coverage of the basic concepts and theory, it explores current trends, results, and applications of binomial ideals to other areas of mathematics. The book begins with a brief, self-contained overview of the modern theory of Gröbner bases and the necessary algebraic and homological concepts from commutative algebra. Binomials and binomial ideals are then considered in detail, along with a short introduction to convex polytopes. Chapters in the remainder of the text can be read independently and explore specific aspects of the theory of binomial ideals, including edge rings and edge polytopes, join-meet ideals of finite lattices, binomial edge ideals, ideals generated by 2-minors, and binomial ideals arising from statistics. Each chapter concludes with a set of exercises and a list of related topics and results that will complement and offer a better understanding of the material presented. Binomial Ideals is suitable for graduate students in courses on commutative algebra, algebraic combinatorics, and statistics. Additionally, researchers interested in any of these areas but familiar with only the basic facts of commutative algebra will find it to be a valuable resource.