Determinants Gröbner Bases And Cohomology

Author : Winfried Bruns,Aldo Conca,Claudiu Raicu,Matteo Varbaro
Publisher : Springer Nature
Page : 514 pages
File Size : 54,8 Mb
Release : 2022-12-02
Category : Mathematics
ISBN : 9783031054808

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Determinants, Gröbner Bases and Cohomology by Winfried Bruns,Aldo Conca,Claudiu Raicu,Matteo Varbaro Pdf

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.

Author : David J. Green
Publisher : Springer Science & Business Media
Page : 156 pages
File Size : 44,5 Mb
Release : 2003-11-18
Category : Mathematics
ISBN : 3540203397

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Gröbner Bases and the Computation of Group Cohomology by David J. Green Pdf

This monograph develops the Gröbner basis methods needed to perform efficient state of the art calculations in the cohomology of finite groups. Results obtained include the first counterexample to the conjecture that the ideal of essential classes squares to zero. The context is J. F. Carlson’s minimal resolutions approach to cohomology computations.

Author : David J. Green
Publisher : Unknown
Page : 156 pages
File Size : 44,9 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 3662191172

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Grobner Bases and the Computation of Group Cohomology by David J. Green Pdf

Author : Takayuki Hibi
Publisher : Springer Science & Business Media
Page : 488 pages
File Size : 47,7 Mb
Release : 2014-01-07
Category : Mathematics
ISBN : 9784431545743

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Gröbner Bases by Takayuki Hibi Pdf

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.

Author : Winfried Bruns,Udo Vetter
Publisher : Springer
Page : 246 pages
File Size : 49,6 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540392743

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Determinantal Rings by Winfried Bruns,Udo Vetter Pdf

Determinantal rings and varieties have been a central topic of commutative algebra and algebraic geometry. Their study has attracted many prominent researchers and has motivated the creation of theories which may now be considered part of general commutative ring theory. The book gives a first coherent treatment of the structure of determinantal rings. The main approach is via the theory of algebras with straightening law. This approach suggest (and is simplified by) the simultaneous treatment of the Schubert subvarieties of Grassmannian. Other methods have not been neglected, however. Principal radical systems are discussed in detail, and one section is devoted to each of invariant and representation theory. While the book is primarily a research monograph, it serves also as a reference source and the reader requires only the basics of commutative algebra together with some supplementary material found in the appendix. The text may be useful for seminars following a course in commutative ring theory since a vast number of notions, results, and techniques can be illustrated significantly by applying them to determinantal rings.

Author : Anonim
Publisher : Unknown
Page : 940 pages
File Size : 40,7 Mb
Release : 2006
Category : Mathematics
ISBN : UOM:39015065183553

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Mathematical Reviews by Anonim Pdf

Author : Irena Peeva
Publisher : Springer Nature
Page : 898 pages
File Size : 47,9 Mb
Release : 2022-02-18
Category : Mathematics
ISBN : 9783030896942

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Commutative Algebra by Irena Peeva Pdf

This contributed volume is a follow-up to the 2013 volume of the same title, published in honor of noted Algebraist David Eisenbud's 65th birthday. It brings together the highest quality expository papers written by leaders and talented junior mathematicians in the field of Commutative Algebra. Contributions cover a very wide range of topics, including core areas in Commutative Algebra and also relations to Algebraic Geometry, Category Theory, Combinatorics, Computational Algebra, Homological Algebra, Hyperplane Arrangements, and Non-commutative Algebra. The book aims to showcase the area and aid junior mathematicians and researchers who are new to the field in broadening their background and gaining a deeper understanding of the current research in this area. Exciting developments are surveyed and many open problems are discussed with the aspiration to inspire the readers and foster further research.

Author : J. M. Landsberg
Publisher : American Mathematical Soc.
Page : 464 pages
File Size : 49,7 Mb
Release : 2011-12-14
Category : Mathematics
ISBN : 9780821869079

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Tensors: Geometry and Applications by J. M. Landsberg Pdf

Tensors are ubiquitous in the sciences. The geometry of tensors is both a powerful tool for extracting information from data sets, and a beautiful subject in its own right. This book has three intended uses: a classroom textbook, a reference work for researchers in the sciences, and an account of classical and modern results in (aspects of) the theory that will be of interest to researchers in geometry. For classroom use, there is a modern introduction to multilinear algebra and to the geometry and representation theory needed to study tensors, including a large number of exercises. For researchers in the sciences, there is information on tensors in table format for easy reference and a summary of the state of the art in elementary language. This is the first book containing many classical results regarding tensors. Particular applications treated in the book include the complexity of matrix multiplication, P versus NP, signal processing, phylogenetics, and algebraic statistics. For geometers, there is material on secant varieties, G-varieties, spaces with finitely many orbits and how these objects arise in applications, discussions of numerous open questions in geometry arising in applications, and expositions of advanced topics such as the proof of the Alexander-Hirschowitz theorem and of the Weyman-Kempf method for computing syzygies.

Author : David Eisenbud,Daniel R. Grayson,Mike Stillman,Bernd Sturmfels
Publisher : Springer Science & Business Media
Page : 354 pages
File Size : 40,5 Mb
Release : 2001-09-25
Category : Mathematics
ISBN : 3540422307

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Computations in Algebraic Geometry with Macaulay 2 by David Eisenbud,Daniel R. Grayson,Mike Stillman,Bernd Sturmfels Pdf

This book presents algorithmic tools for algebraic geometry, with experimental applications. It also introduces Macaulay 2, a computer algebra system supporting research in algebraic geometry, commutative algebra, and their applications. The algorithmic tools presented here are designed to serve readers wishing to bring such tools to bear on their own problems. The first part of the book covers Macaulay 2 using concrete applications; the second emphasizes details of the mathematics.

Author : Shum Kar Ping,Zelmanov Efim,Kolesnikov Pavel,Wong Anita S M
Publisher : World Scientific
Page : 500 pages
File Size : 55,8 Mb
Release : 2020-02-18
Category : Mathematics
ISBN : 9789811215483

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New Trends In Algebras And Combinatorics - Proceedings Of The Third International Congress In Algebras And Combinatorics (Icac2017) by Shum Kar Ping,Zelmanov Efim,Kolesnikov Pavel,Wong Anita S M Pdf

This volume composed of twenty four research articles which are selected from the keynote speakers and invited lectures presented in the 3rd International Congress in Algebra and Combinatorics (ICAC2017) held on 25-28 August 2017 in Hong Kong and one additional invited article. This congress was specially dedicated to Professor Leonid Bokut on the occasion of his 80th birthday.

Author : Anthony W. Knapp
Publisher : Springer Science & Business Media
Page : 730 pages
File Size : 48,9 Mb
Release : 2007-10-11
Category : Mathematics
ISBN : 9780817646134

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Advanced Algebra by Anthony W. Knapp Pdf

Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Advanced Algebra includes chapters on modern algebra which treat various topics in commutative and noncommutative algebra and provide introductions to the theory of associative algebras, homological algebras, algebraic number theory, and algebraic geometry. Many examples and hundreds of problems are included, along with hints or complete solutions for most of the problems. Together the two books give the reader a global view of algebra and its role in mathematics as a whole.

Author : Ezra Miller,Bernd Sturmfels
Publisher : Springer Science & Business Media
Page : 442 pages
File Size : 47,5 Mb
Release : 2005-06-21
Category : Mathematics
ISBN : 0387237070

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Combinatorial Commutative Algebra by Ezra Miller,Bernd Sturmfels Pdf

Recent developments are covered Contains over 100 figures and 250 exercises Includes complete proofs

Author : Bernd Sturmfels
Publisher : American Mathematical Soc.
Page : 176 pages
File Size : 52,6 Mb
Release : 1996
Category : Mathematics
ISBN : 9780821804872

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Grobner Bases and Convex Polytopes by Bernd Sturmfels Pdf

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.

Author : Valeri? Valer?evich Dolotin,A. Morozov,Al?bert Dmitrievich Morozov
Publisher : World Scientific
Page : 286 pages
File Size : 40,7 Mb
Release : 2007
Category : Mathematics
ISBN : 9789812708007

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Introduction to Non-linear Algebra by Valeri? Valer?evich Dolotin,A. Morozov,Al?bert Dmitrievich Morozov Pdf

Literaturverz. S. 267 - 269

Author : Murray R. Bremner,Vladimir Dotsenko
Publisher : CRC Press
Page : 382 pages
File Size : 44,6 Mb
Release : 2016-04-06
Category : Mathematics
ISBN : 9781482248579

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Algebraic Operads by Murray R. Bremner,Vladimir Dotsenko Pdf

Algebraic Operads: An Algorithmic Companion presents a systematic treatment of Grobner bases in several contexts. The book builds up to the theory of Grobner bases for operads due to the second author and Khoroshkin as well as various applications of the corresponding diamond lemmas in algebra. The authors present a variety of topics including: non