Grobner Bases In Ring Theory

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Grobner Bases in Ring Theory

Huishi Li
Grobner Bases in Ring Theory Book PDF

Author : Huishi Li
Publisher : World Scientific
Page : 295 pages
File Size : 42,6 Mb
Release : 2012
Category : Mathematics
ISBN : 9789814365147

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Grobner Bases in Ring Theory by Huishi Li Pdf

1. Preliminaries. 1.1. Presenting algebras by relations. 1.2. S-graded algebras and modules. 1.3. [symbol]-filtered algebras and modules -- 2. The [symbol]-leading homogeneous algebra A[symbol]. 2.1. Recognizing A via G[symbol](A): part 1. 2.2. Recognizing A via G[symbol](A): part 2. 2.3. The [symbol-graded isomorphism A[symbol](A). 2.4. Recognizing A via A[symbol] -- 3. Grobner bases: conception and construction. 3.1. Monomial ordering and admissible system. 3.2. Division algorithm and Grobner basis. 3.3. Grobner bases and normal elements. 3.4. Grobner bases w.r.t. skew multiplicative K-bases. 3.5. Grobner bases in K[symbol] and KQ. 3.6. (De)homogenized Grobner bases. 3.7. dh-closed homogeneous Grobner bases -- 4. Grobner basis theory meets PBW theory. 4.1. [symbol]-standard basis [symbol]-PBW isomorphism. 4.2. Realizing [symbol]-PBW isomorphism by Grobner basis. 4.3. Classical PBW K-bases vs Grobner bases. 4.4. Solvable polynomial algebras revisited -- 5. Using A[symbol] in terms of Grobner bases. 5.1. The working strategy. 5.2. Ufnarovski graph. 5.3. Determination of Gelfand-Kirillov Dimension. 5.4. Recognizing Noetherianity. 5.5. Recognizing (semi- )primeness and PI-property. 5.6. Anick's resolution over monomial algebras. 5.7. Recognizing finiteness of global dimension. 5.8. Determination of Hilbert series -- 6. Recognizing (non- )homogeneous p-Koszulity via A[symbol]. 6.1. (Non- )homogeneous p-Koszul algebras. 6.2. Anick's resolution and homogeneous p-Koszulity. 6.3. Working in terms of Grobner bases -- 7. A study of Rees algebra by Grobner bases. 7.1. Defining [symbol] by [symbol]. 7.2. Defining [symbol] by [symbol]. 7.3. Recognizing structural properties of [symbol] via [symbol]. 7.4. An application to regular central extensions. 7.5. Algebras defined by dh-closed homogeneous Grobner bases -- 8. Looking for more Grobner bases. 8.1. Lifting (finite) Grobner bases from O[symbol]. 8.2. Lifting (finite) Grobner bases from a class of algebras. 8.3. New examples of Grobner basis theory. 8.4. Skew 2-nomial algebras. 8.5. Almost skew 2-nomial algebras

Gr”bner Bases in Ring Theory

Huishi Li
Gr bner Bases in Ring Theory Book PDF

Author : Huishi Li
Publisher : World Scientific
Page : 295 pages
File Size : 50,7 Mb
Release : 2012
Category : Mathematics
ISBN : 9789814365130

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Gr”bner Bases in Ring Theory by Huishi Li Pdf

This monograph strives to introduce a solid foundation on the usage of Gr”bner bases in ring theory by focusing on noncommutative associative algebras defined by relations over a field K. It also reveals the intrinsic structural properties of Gr”bner bases, presents a constructive PBW theory in a quite extensive context and, along the routes built via the PBW theory, the book demonstrates novel methods of using Gr”bner bases in determining and recognizing many more structural properties of algebras, such as the Gelfand?Kirillov dimension, Noetherianity, (semi-)primeness, PI-property, finiteness of global homological dimension, Hilbert series, (non-)homogeneous p-Koszulity, PBW-deformation, and regular central extension.With a self-contained and constructive Gr”bner basis theory for algebras with a skew multiplicative K-basis, numerous illuminating examples are constructed in the book for illustrating and extending the topics studied. Moreover, perspectives of further study on the topics are prompted at appropriate points. This book can be of considerable interest to researchers and graduate students in computational (computer) algebra, computational (noncommutative) algebraic geometry; especially for those working on the structure theory of rings, algebras and their modules (representations).

An Introduction to Grobner Bases

William W. Adams and Philippe Loustaunau
An Introduction to Grobner Bases Book PDF

Author : William W. Adams and Philippe Loustaunau
Publisher : American Mathematical Soc.
Page : 308 pages
File Size : 45,8 Mb
Release : 1994-07-21
Category : Mathematics
ISBN : 0821872168

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An Introduction to Grobner Bases by William W. Adams and Philippe Loustaunau Pdf

A very carefully crafted introduction to the theory and some of the applications of Grobner bases ... contains a wealth of illustrative examples and a wide variety of useful exercises, the discussion is everywhere well-motivated, and further developments and important issues are well sign-posted ... has many solid virtues and is an ideal text for beginners in the subject ... certainly an excellent text. --Bulletin of the London Mathematical Society As the primary tool for doing explicit computations in polynomial rings in many variables, Grobner bases are an important component of all computer algebra systems. They are also important in computational commutative algebra and algebraic geometry. This book provides a leisurely and fairly comprehensive introduction to Grobner bases and their applications. Adams and Loustaunau cover the following topics: the theory and construction of Grobner bases for polynomials with coefficients in a field, applications of Grobner bases to computational problems involving rings of polynomials in many variables, a method for computing syzygy modules and Grobner bases in modules, and the theory of Grobner bases for polynomials with coefficients in rings. With over 120 worked-out examples and 200 exercises, this book is aimed at advanced undergraduate and graduate students. It would be suitable as a supplement to a course in commutative algebra or as a textbook for a course in computer algebra or computational commutative algebra. This book would also be appropriate for students of computer science and engineering who have some acquaintance with modern algebra.

Gröbner Bases and Applications

Bruno Buchberger,Franz Winkler
Gr bner Bases and Applications Book PDF

Author : Bruno Buchberger,Franz Winkler
Publisher : Cambridge University Press
Page : 566 pages
File Size : 42,5 Mb
Release : 1998-02-26
Category : Mathematics
ISBN : 0521632986

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Gröbner Bases and Applications by Bruno Buchberger,Franz Winkler Pdf

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

Gröbner Bases, Coding, and Cryptography

Massimiliano Sala,Teo Mora,Ludovic Perret,Shojiro Sakata,Carlo Traverso
Gr bner Bases Coding and Cryptography Book PDF

Author : Massimiliano Sala,Teo Mora,Ludovic Perret,Shojiro Sakata,Carlo Traverso
Publisher : Springer Science & Business Media
Page : 428 pages
File Size : 55,7 Mb
Release : 2009-05-28
Category : Mathematics
ISBN : 9783540938064

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Gröbner Bases, Coding, and Cryptography by Massimiliano Sala,Teo Mora,Ludovic Perret,Shojiro Sakata,Carlo Traverso Pdf

Coding theory and cryptography allow secure and reliable data transmission, which is at the heart of modern communication. Nowadays, it is hard to find an electronic device without some code inside. Gröbner bases have emerged as the main tool in computational algebra, permitting numerous applications, both in theoretical contexts and in practical situations. This book is the first book ever giving a comprehensive overview on the application of commutative algebra to coding theory and cryptography. For example, all important properties of algebraic/geometric coding systems (including encoding, construction, decoding, list decoding) are individually analysed, reporting all significant approaches appeared in the literature. Also, stream ciphers, PK cryptography, symmetric cryptography and Polly Cracker systems deserve each a separate chapter, where all the relevant literature is reported and compared. While many short notes hint at new exciting directions, the reader will find that all chapters fit nicely within a unified notation.

Ring Theory And Algebraic Geometry

A. Granja,J.A. Hermida Alonso,A Verschoren
Ring Theory And Algebraic Geometry Book PDF

Author : A. Granja,J.A. Hermida Alonso,A Verschoren
Publisher : CRC Press
Page : 363 pages
File Size : 47,7 Mb
Release : 2001-05-08
Category : Mathematics
ISBN : 9780203907962

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Ring Theory And Algebraic Geometry by A. Granja,J.A. Hermida Alonso,A Verschoren Pdf

Focuses on the interaction between algebra and algebraic geometry, including high-level research papers and surveys contributed by over 40 top specialists representing more than 15 countries worldwide. Describes abelian groups and lattices, algebras and binomial ideals, cones and fans, affine and projective algebraic varieties, simplicial and cellular complexes, polytopes, and arithmetics.

Constructive Commutative Algebra

Ihsen Yengui
Constructive Commutative Algebra Book PDF

Author : Ihsen Yengui
Publisher : Springer
Page : 277 pages
File Size : 46,5 Mb
Release : 2015-12-11
Category : Mathematics
ISBN : 9783319194943

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Constructive Commutative Algebra by Ihsen Yengui Pdf

The main goal of this book is to find the constructive content hidden in abstract proofs of concrete theorems in Commutative Algebra, especially in well-known theorems concerning projective modules over polynomial rings (mainly the Quillen-Suslin theorem) and syzygies of multivariate polynomials with coefficients in a valuation ring. Simple and constructive proofs of some results in the theory of projective modules over polynomial rings are also given, and light is cast upon recent progress on the Hermite ring and Gröbner ring conjectures. New conjectures on unimodular completion arising from our constructive approach to the unimodular completion problem are presented. Constructive algebra can be understood as a first preprocessing step for computer algebra that leads to the discovery of general algorithms, even if they are sometimes not efficient. From a logical point of view, the dynamical evaluation gives a constructive substitute for two highly nonconstructive tools of abstract algebra: the Law of Excluded Middle and Zorn's Lemma. For instance, these tools are required in order to construct the complete prime factorization of an ideal in a Dedekind ring, whereas the dynamical method reveals the computational content of this construction. These lecture notes follow this dynamical philosophy.

Grobner Bases in Commutative Algebra

Viviana Ene,JŸrgen Herzog
Grobner Bases in Commutative Algebra Book PDF

Author : Viviana Ene,JŸrgen Herzog
Publisher : American Mathematical Soc.
Page : 178 pages
File Size : 53,7 Mb
Release : 2011-12-01
Category : Mathematics
ISBN : 9780821872871

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Grobner Bases in Commutative Algebra by Viviana Ene,JŸrgen Herzog Pdf

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.

Gröbner Bases and Convex Polytopes

Bernd Sturmfels
Gr bner Bases and Convex Polytopes Book PDF

Author : Bernd Sturmfels
Publisher : American Mathematical Soc.
Page : 162 pages
File Size : 42,5 Mb
Release : 1996
Category : Mathematics
ISBN : 9780821804872

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Gröbner 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 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.

Computational Methods for Representations of Groups and Algebras

P. Dräxler,G. Michler,C.M. Ringel
Computational Methods for Representations of Groups and Algebras Book PDF

Author : P. Dräxler,G. Michler,C.M. Ringel
Publisher : Birkhäuser
Page : 358 pages
File Size : 52,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034887168

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Computational Methods for Representations of Groups and Algebras by P. Dräxler,G. Michler,C.M. Ringel Pdf

This book presents material from 3 survey lectures and 14 additional invited lectures given at the Euroconference "Computational Methods for Representations of Groups and Algebras" held at Essen University in April 1997. The purpose of this meeting was to provide a survey of general theoretical and computational methods and recent advances in the representation theory of groups and algebras. The foundations of these research areas were laid in survey articles by P. Dräxler and R. Nörenberg on "Classification problems in the representation theory of finite-dimensional algebras", R. A. Wilson on "Construction of finite matrix groups" and E. Green on "Noncommutative Gröbner bases, and projective resolutions". Furthermore, new applications of the computational methods in linear algebra to the revision of the classification of finite simple sporadic groups are presented. Computational tools (including high-performance computations on supercomputers) have become increasingly important for classification problems. They are also inevitable for the construction of projective resolutions of finitely generated modules over finite-dimensional algebras and the study of group cohomology and rings of invariants. A major part of this book is devoted to a survey of algorithms for computing special examples in the study of Grothendieck groups, quadratic forms and derived categories of finite-dimensional algebras. Open questions on Lie algebras, Bruhat orders, Coxeter groups and Kazhdan Lusztig polynomials are investigated with the aid of computer programs. The contents of this book provide an overview on the present state of the art. Therefore it will be very useful for graduate students and researchers in mathematics, computer science and physics.

An Introduction to Gröbner Bases

William Wells Adams,Philippe Loustaunau
An Introduction to Gr bner Bases Book PDF

Author : William Wells Adams,Philippe Loustaunau
Publisher : American Mathematical Soc.
Page : 289 pages
File Size : 50,9 Mb
Release : 1994
Category : Mathematics
ISBN : 9780821838044

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An Introduction to Gröbner Bases by William Wells Adams,Philippe Loustaunau Pdf

A very carefully crafted introduction to the theory and some of the applications of Grobner bases ... contains a wealth of illustrative examples and a wide variety of useful exercises, the discussion is everywhere well-motivated, and further developments and important issues are well sign-posted ... has many solid virtues and is an ideal text for beginners in the subject ... certainly an excellent text. --Bulletin of the London Mathematical Society As the primary tool for doing explicit computations in polynomial rings in many variables, Grobner bases are an important component of all computer algebra systems. They are also important in computational commutative algebra and algebraic geometry. This book provides a leisurely and fairly comprehensive introduction to Grobner bases and their applications. Adams and Loustaunau cover the following topics: the theory and construction of Grobner bases for polynomials with coefficients in a field, applications of Grobner bases to computational problems involving rings of polynomials in many variables, a method for computing syzygy modules and Grobner bases in modules, and the theory of Grobner bases for polynomials with coefficients in rings. With over 120 worked-out examples and 200 exercises, this book is aimed at advanced undergraduate and graduate students. It would be suitable as a supplement to a course in commutative algebra or as a textbook for a course in computer algebra or computational commutative algebra. This book would also be appropriate for students of computer science and engineering who have some acquaintance with modern algebra.

Gröbner Bases and the Computation of Group Cohomology

David J. Green
Gr bner Bases and the Computation of Group Cohomology Book PDF

Author : David J. Green
Publisher : Springer Science & Business Media
Page : 156 pages
File Size : 53,5 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.

Boolean Gröbner Bases

Michael Brickenstein
Boolean Gr bner Bases Book PDF

Author : Michael Brickenstein
Publisher : Logos Verlag Berlin GmbH
Page : 158 pages
File Size : 53,8 Mb
Release : 2010
Category : Computers
ISBN : 9783832525972

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Boolean Gröbner Bases by Michael Brickenstein Pdf

There exist very few concepts in computational algebra which are as central to theory and applications as Grobner bases. This thesis describes theory, algorithms and applications for the special case of Boolean polynomials. These parts form the mathematical foundations of the PolyBoRi framework (developed by the author together with Alexander Dreyer). The PolyBoRi framework has applications spread over a large number of domains ranging from formal verification, computational biology to cryptanalysis and many more. It is emerged to a worldwide audience by the Sage computational algebra system.

Skew PBW Extensions

William Fajardo,Claudia Gallego,Oswaldo Lezama,Armando Reyes,Héctor Suárez,Helbert Venegas
Skew PBW Extensions Book PDF

Author : William Fajardo,Claudia Gallego,Oswaldo Lezama,Armando Reyes,Héctor Suárez,Helbert Venegas
Publisher : Springer Nature
Page : 584 pages
File Size : 49,9 Mb
Release : 2020-12-11
Category : Mathematics
ISBN : 9783030533786

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Skew PBW Extensions by William Fajardo,Claudia Gallego,Oswaldo Lezama,Armando Reyes,Héctor Suárez,Helbert Venegas Pdf

This monograph is devoted to a new class of non-commutative rings, skew Poincaré–Birkhoff–Witt (PBW) extensions. Beginning with the basic definitions and ring-module theoretic/homological properties, it goes on to investigate finitely generated projective modules over skew PBW extensions from a matrix point of view. To make this theory constructive, the theory of Gröbner bases of left (right) ideals and modules for bijective skew PBW extensions is developed. For example, syzygies and the Ext and Tor modules over these rings are computed. Finally, applications to some key topics in the noncommutative algebraic geometry of quantum algebras are given, including an investigation of semi-graded Koszul algebras and semi-graded Artin–Schelter regular algebras, and the noncommutative Zariski cancellation problem. The book is addressed to researchers in noncommutative algebra and algebraic geometry as well as to graduate students and advanced undergraduate students.

Two Algebraic Byways from Differential Equations: Gröbner Bases and Quivers

Kenji Iohara,Philippe Malbos,Masa-Hiko Saito,Nobuki Takayama
Two Algebraic Byways from Differential Equations Gr bner Bases and Quivers Book PDF

Author : Kenji Iohara,Philippe Malbos,Masa-Hiko Saito,Nobuki Takayama
Publisher : Springer Nature
Page : 375 pages
File Size : 50,7 Mb
Release : 2020-02-20
Category : Mathematics
ISBN : 9783030264543

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Two Algebraic Byways from Differential Equations: Gröbner Bases and Quivers by Kenji Iohara,Philippe Malbos,Masa-Hiko Saito,Nobuki Takayama Pdf

This edited volume presents a fascinating collection of lecture notes focusing on differential equations from two viewpoints: formal calculus (through the theory of Gröbner bases) and geometry (via quiver theory). Gröbner bases serve as effective models for computation in algebras of various types. Although the theory of Gröbner bases was developed in the second half of the 20th century, many works on computational methods in algebra were published well before the introduction of the modern algebraic language. Since then, new algorithms have been developed and the theory itself has greatly expanded. In comparison, diagrammatic methods in representation theory are relatively new, with the quiver varieties only being introduced – with big impact – in the 1990s. Divided into two parts, the book first discusses the theory of Gröbner bases in their commutative and noncommutative contexts, with a focus on algorithmic aspects and applications of Gröbner bases to analysis on systems of partial differential equations, effective analysis on rings of differential operators, and homological algebra. It then introduces representations of quivers, quiver varieties and their applications to the moduli spaces of meromorphic connections on the complex projective line. While no particular reader background is assumed, the book is intended for graduate students in mathematics, engineering and related fields, as well as researchers and scholars.