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Grobner Bases And Convex Polytopes
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Book Synopsis Gröbner Bases and Convex Polytopes by : Bernd Sturmfels
Download or read book Gröbner Bases and Convex Polytopes written by Bernd Sturmfels and published by American Mathematical Soc.. This book was released on 1996 with total page 162 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 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.
Book Synopsis Gröbner Bases and Convex Polytopes by : Bernd Sturmfels
Download or read book Gröbner Bases and Convex Polytopes written by Bernd Sturmfels and published by American Mathematical Soc.. This book was released on 1996 with total page 162 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Gröbner Bases written by Takayuki Hibi and published by Springer Science & Business Media. This book was released on 2014-01-07 with total page 488 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Minkowski Addition of Polytopes by : Peter Gritzmann
Download or read book Minkowski Addition of Polytopes written by Peter Gritzmann and published by . This book was released on 1990 with total page 62 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Convex Polytopes written by P. McMullen and published by CUP Archive. This book was released on 1971-07-02 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Harmony of Gröbner Bases and the Modern Industrial Society by : Takayuki Hibi
Download or read book Harmony of Gröbner Bases and the Modern Industrial Society written by Takayuki Hibi and published by World Scientific. This book was released on 2012 with total page 385 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Harmony of Gröbner Bases and the Modern Industrial Society by : Takayuki Hibi
Download or read book Harmony of Gröbner Bases and the Modern Industrial Society written by Takayuki Hibi and published by World Scientific. This book was released on 2012-03-21 with total page 388 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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. Contents:Multidegree for Bifiltered D-modules and Hypergeometric Systems (R Arcadias)Desingularization Algorithms: A Comparison from the Practical Point of View (R Blanco and A Frühbis-Krüger)Computing Localizations Iteratively (F J Castro-Jiménez and A Leykin)KNOPPIX/Math: A Live System for Mathematics (T Hamada and KNOPPIX/Math Committers)Running Markov Chain without Markov Basis (H Hara, S Aoki and A Takemura)Degree Bounds for a Minimal Markov Basis for the Three-state Toric Homogeneous Markov Chain Model (D Haws, A Martín del Campo and R Yoshida)First Steps toward the Geometry of Cophylogeny (P Huggins, M Owen and R Yoshida)Cones of Elementary Imsets and Supermodular Functions: A Review and Some New Results (T Kashimura, T Sei, A Takemura and K Tanaka)Non-vanishingness of Betti Numbers of Edge Ideals (K Kimura)Abstract Tubes Associated with Perturbed Polyhedra with Applications to Multidimensional Normal Probability Computations (S Kuriki, T Miwa and A J Hayter)An Algorithm of Computing Inhomogeneous Difference Equations for a Definite Sum (H Nakayama)Incomplete A-Hypergeometric Systems (K Nishiyama and N Takayama)Implementation of a Primary Decomposition Package (M Noro)On Computation of the Characteristic Polynomials of the Discriminantal Arrangements and the Arrangements Generated by Generic Points (Y Numata and A Takemura)A Dictionary of Gröbner Bases of Toric Ideals (H Ohsugi)Log-linear Model Estimation for Stratified Educational Data (T Otsu)Toric Statistical Models: Ising and Markov (G Pistone and M P Rogantin)Algebraic Reliability Based on Monomial Ideals: A Review (E Sáenz-de-Cabezón and H P Wynn)On Irreducibility of Algebroid Curves over the Complex Number Field (T Shibuta)Polyhedral Approach to Statistical Learning Graphical Models (M Studený, D Haws, R Hemmecke and S Lindner) Readership: Graduates and researchers in the field of Gröbner bases. Keywords:Gröbner Basis;Algebraic Statistics;D-Module;Computational Algebra;AlgorithmKey Features:Comprehensive treatment of Gröbner basesArticles by leading figures in the mathematics worldA guidebook for graduate studentsThe reader can see a panoramic view of Gröbner bases
Book Synopsis An Introduction to Convex Polytopes by : Arne Brondsted
Download or read book An Introduction to Convex Polytopes written by Arne Brondsted and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 168 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this book is to introduce the reader to the fascinating world of convex polytopes. The highlights of the book are three main theorems in the combinatorial theory of convex polytopes, known as the Dehn-Sommerville Relations, the Upper Bound Theorem and the Lower Bound Theorem. All the background information on convex sets and convex polytopes which is m~eded to under stand and appreciate these three theorems is developed in detail. This background material also forms a basis for studying other aspects of polytope theory. The Dehn-Sommerville Relations are classical, whereas the proofs of the Upper Bound Theorem and the Lower Bound Theorem are of more recent date: they were found in the early 1970's by P. McMullen and D. Barnette, respectively. A famous conjecture of P. McMullen on the charac terization off-vectors of simplicial or simple polytopes dates from the same period; the book ends with a brief discussion of this conjecture and some of its relations to the Dehn-Sommerville Relations, the Upper Bound Theorem and the Lower Bound Theorem. However, the recent proofs that McMullen's conditions are both sufficient (L. J. Billera and C. W. Lee, 1980) and necessary (R. P. Stanley, 1980) go beyond the scope of the book. Prerequisites for reading the book are modest: standard linear algebra and elementary point set topology in [R1d will suffice.
Book Synopsis Gröbner Bases and Applications by : Bruno Buchberger
Download or read book Gröbner Bases and Applications written by Bruno Buchberger and published by Cambridge University Press. This book was released on 1998-02-26 with total page 566 pages. Available in PDF, EPUB and Kindle. Book excerpt: Comprehensive account of theory and applications of Gröbner bases, co-edited by the subject's inventor.
Book Synopsis Convex Polytopes by : Branko Grünbaum
Download or read book Convex Polytopes written by Branko Grünbaum and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 561 pages. Available in PDF, EPUB and Kindle. Book excerpt: "The original edition [...] inspired a whole generation of grateful workers in polytope theory. Without it, it is doubtful whether many of the subsequent advances in the subject would have been made. The many seeds it sowed have since grown into healthy trees, with vigorous branches and luxuriant foliage. It is good to see it in print once again." --Peter McMullen, University College London
Download or read book Binomial Ideals written by Jürgen Herzog and published by Springer. This book was released on 2018-09-28 with total page 321 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Determinants, Gröbner Bases and Cohomology by : Winfried Bruns
Download or read book Determinants, Gröbner Bases and Cohomology written by Winfried Bruns and published by Springer Nature. This book was released on 2022-12-02 with total page 514 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers an up-to-date, comprehensive account of determinantal rings and varieties, presenting a multitude of methods used in their study, with tools from combinatorics, algebra, representation theory and geometry. After a concise introduction to Gröbner and Sagbi bases, determinantal ideals are studied via the standard monomial theory and the straightening law. This opens the door for representation theoretic methods, such as the Robinson–Schensted–Knuth correspondence, which provide a description of the Gröbner bases of determinantal ideals, yielding homological and enumerative theorems on determinantal rings. Sagbi bases then lead to the introduction of toric methods. In positive characteristic, the Frobenius functor is used to study properties of singularities, such as F-regularity and F-rationality. Castelnuovo–Mumford regularity, an important complexity measure in commutative algebra and algebraic geometry, is introduced in the general setting of a Noetherian base ring and then applied to powers and products of ideals. The remainder of the book focuses on algebraic geometry, where general vanishing results for the cohomology of line bundles on flag varieties are presented and used to obtain asymptotic values of the regularity of symbolic powers of determinantal ideals. In characteristic zero, the Borel–Weil–Bott theorem provides sharper results for GL-invariant ideals. The book concludes with a computation of cohomology with support in determinantal ideals and a survey of their free resolutions. Determinants, Gröbner Bases and Cohomology provides a unique reference for the theory of determinantal ideals and varieties, as well as an introduction to the beautiful mathematics developed in their study. Accessible to graduate students with basic grounding in commutative algebra and algebraic geometry, it can be used alongside general texts to illustrate the theory with a particularly interesting and important class of varieties.
Book Synopsis Grobner Bases in Commutative Algebra by : Viviana Ene
Download or read book Grobner Bases in Commutative Algebra written by Viviana Ene and published by American Mathematical Soc.. This book was released on 2011-12-01 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a concise yet comprehensive and self-contained introduction to Grobner basis theory and its applications to various current research topics in commutative algebra. It especially aims to help young researchers become acquainted with fundamental tools and techniques related to Grobner bases which are used in commutative algebra and to arouse their interest in exploring further topics such as toric rings, Koszul and Rees algebras, determinantal ideal theory, binomial edge ideals, and their applications to statistics. The book can be used for graduate courses and self-study. More than 100 problems will help the readers to better understand the main theoretical results and will inspire them to further investigate the topics studied in this book.
Book Synopsis Lectures on Polytopes by : Günter M. Ziegler
Download or read book Lectures on Polytopes written by Günter M. Ziegler and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 347 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on a graduate course at the Technische Universität, Berlin, these lectures present a wealth of material on the modern theory of convex polytopes. The straightforward exposition features many illustrations, and complete proofs for most theorems. With only linear algebra as a prerequisite, it takes the reader quickly from the basics to topics of recent research. The lectures introduce basic facts about polytopes, with an emphasis on methods that yield the results, discuss important examples and elegant constructions, and show the excitement of current work in the field. They will provide interesting and enjoyable reading for researchers as well as students.
Book Synopsis Lectures in Geometric Combinatorics by : Rekha R. Thomas
Download or read book Lectures in Geometric Combinatorics written by Rekha R. Thomas and published by American Mathematical Soc.. This book was released on 2006 with total page 156 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a course in the geometry of convex polytopes in arbitrary dimension, suitable for an advanced undergraduate or beginning graduate student. The book starts with the basics of polytope theory. Schlegel and Gale diagrams are introduced as geometric tools to visualize polytopes in high dimension and to unearth bizarre phenomena in polytopes. The heart of the book is a treatment of the secondary polytope of a point configuration and its connections to the statepolytope of the toric ideal defined by the configuration. These polytopes are relatively recent constructs with numerous connections to discrete geometry, classical algebraic geometry, symplectic geometry, and combinatorics. The connections rely on Grobner bases of toric ideals and other methods fromcommutative algebra. The book is self-contained and does not require any background beyond basic linear algebra. With numerous figures and exercises, it can be used as a textbook for courses on geometric, combinatorial, and computational aspects of the theory of polytopes.
Book Synopsis Handbook of Discrete and Computational Geometry, Second Edition by : Csaba D. Toth
Download or read book Handbook of Discrete and Computational Geometry, Second Edition written by Csaba D. Toth and published by CRC Press. This book was released on 2004-04-13 with total page 1557 pages. Available in PDF, EPUB and Kindle. Book excerpt: While high-quality books and journals in this field continue to proliferate, none has yet come close to matching the Handbook of Discrete and Computational Geometry, which in its first edition, quickly became the definitive reference work in its field. But with the rapid growth of the discipline and the many advances made over the past seven years, it's time to bring this standard-setting reference up to date. Editors Jacob E. Goodman and Joseph O'Rourke reassembled their stellar panel of contributors, added manymore, and together thoroughly revised their work to make the most important results and methods, both classic and cutting-edge, accessible in one convenient volume. Now over more then 1500 pages, the Handbook of Discrete and Computational Geometry, Second Edition once again provides unparalleled, authoritative coverage of theory, methods, and applications. Highlights of the Second Edition: Thirteen new chapters: Five on applications and others on collision detection, nearest neighbors in high-dimensional spaces, curve and surface reconstruction, embeddings of finite metric spaces, polygonal linkages, the discrepancy method, and geometric graph theory Thorough revisions of all remaining chapters Extended coverage of computational geometry software, now comprising two chapters: one on the LEDA and CGAL libraries, the other on additional software Two indices: An Index of Defined Terms and an Index of Cited Authors Greatly expanded bibliographies
Author :David A. Cox Dinesh N. Manocha Bernd Sturmfels Publisher :American Mathematical Soc. ISBN 13 :9780821867587 Total Pages :194 pages Book Rating :4.8/5 (675 download)
Book Synopsis Applications of Computational Algebraic Geometry by : David A. Cox Dinesh N. Manocha Bernd Sturmfels
Download or read book Applications of Computational Algebraic Geometry written by David A. Cox Dinesh N. Manocha Bernd Sturmfels and published by American Mathematical Soc.. This book was released on with total page 194 pages. Available in PDF, EPUB and Kindle. Book excerpt:
