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This book demonstrates current trends in research on combinatorial and computational commutative algebra with a primary emphasis on topics related to monomial ideals. Providing a useful and quick introduction to areas of research spanning these fields, Monomial Ideals is split into three parts. Part I offers a quick introduction to the modern theory of Gröbner bases as well as the detailed study of generic initial ideals. Part II supplies Hilbert functions and resolutions and some of the combinatorics related to monomial ideals including the Kruskal—Katona theorem and algebraic aspects of Alexander duality. Part III discusses combinatorial applications of monomial ideals, providing a valuable overview of some of the central trends in algebraic combinatorics. Main subjects include edge ideals of finite graphs, powers of ideals, algebraic shifting theory and an introduction to discrete polymatroids. Theory is complemented by a number of examples and exercises throughout, bringing the reader to a deeper understanding of concepts explored within the text. Self-contained and concise, this book will appeal to a wide range of readers, including PhD students on advanced courses, experienced researchers, and combinatorialists and non-specialists with a basic knowledge of commutative algebra. Since their first meeting in 1985, Juergen Herzog (Universität Duisburg-Essen, Germany) and Takayuki Hibi (Osaka University, Japan), have worked together on a number of research projects, of which recent results are presented in this monograph.

Algebra. --- Mathematics. --- Commutative algebra --- Combinatorial analysis --- Grèobner bases --- Characteristic functions --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Commutative algebra. --- Commutative rings. --- Commutative Rings and Algebras. --- Mathematical analysis --- Characteristic functions. --- Combinatorial analysis. --- Gröbner bases. --- Rings (Algebra)

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This book is divided into two parts, one theoretical and one focusing on applications, and offers a complete description of the Canonical Gröbner Cover, the most accurate algebraic method for discussing parametric polynomial systems. It also includes applications to the Automatic Deduction of Geometric Theorems, Loci Computation and Envelopes. The theoretical part is a self-contained exposition on the theory of Parametric Gröbner Systems and Bases. It begins with Weispfenning’s introduction of Comprehensive Gröbner Systems (CGS) in 1992, and provides a complete description of the Gröbner Cover (GC), which includes a canonical discussion of a set of parametric polynomial equations developed by Michael Wibmer and the author. In turn, the application part selects three problems for which the Gröbner Cover offers valuable new perspectives. The automatic deduction of geometric theorems (ADGT) becomes fully automatic and straightforward using GC, representing a major improvement on all previous methods. In terms of loci and envelope computation, GC makes it possible to introduce a taxonomy of the components and automatically compute it. The book also generalizes the definition of the envelope of a family of hypersurfaces, and provides algorithms for its computation, as well as for discussing how to determine the real envelope. All the algorithms described here have also been included in the software library “grobcov.lib” implemented in Singular by the author, and serve as a User Manual for it.

Gröbner bases. --- Gröbner basis theory --- Commutative algebra --- Algebra. --- Field theory (Physics). --- Algebra --- Commutative Rings and Algebras. --- Field Theory and Polynomials. --- Symbolic and Algebraic Manipulation. --- Data processing. --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics --- Mathematics --- Mathematical analysis --- Commutative algebra. --- Commutative rings. --- Computer science—Mathematics. --- Rings (Algebra)

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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.

Coding theory. --- Cryptography. --- Gro ̈bner bases. --- Grèobner bases --- Coding theory --- Cryptography --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Gröbner bases. --- Gröbner basis theory --- Cryptanalysis --- Cryptology --- Secret writing --- Steganography --- Mathematics. --- Data encryption (Computer science). --- Computers. --- Computer science --- Algebra. --- Discrete mathematics. --- Combinatorics. --- Discrete Mathematics. --- Data Encryption. --- Mathematics of Computing. --- Theory of Computation. --- Commutative algebra --- Signs and symbols --- Symbolism --- Writing --- Ciphers --- Data encryption (Computer science) --- Data compression (Telecommunication) --- Digital electronics --- Information theory --- Machine theory --- Signal theory (Telecommunication) --- Computer programming --- Computer science. --- Information theory. --- Cryptology. --- Communication theory --- Communication --- Cybernetics --- Informatics --- Science --- Data encoding (Computer science) --- Encryption of data (Computer science) --- Computer security --- Combinatorics --- Mathematical analysis --- Computer science—Mathematics. --- Discrete mathematical structures --- Mathematical structures, Discrete --- Structures, Discrete mathematical --- Numerical analysis --- Automatic computers --- Automatic data processors --- Computer hardware --- Computing machines (Computers) --- Electronic brains --- Electronic calculating-machines --- Electronic computers --- Hardware, Computer --- Computer systems --- Calculators --- Cyberspace --- Grobner bases.

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Gröbner bases --- ALGEBRA --- data processing --- Grobner bases. --- Grobner bases --- Gröbner basis theory --- Bases de Gröbner --- Gröbner bases --- Bases de Gröbner --- Algebra --- 512.55 --- 512.55 Rings and modules --- Rings and modules --- Commutative algebra --- Data processing --- Ordered algebraic structures --- Data processing. --- Algèbre --- Informatique --- Geometry, Algebraic --- Géométrie algébrique. --- Algorithms --- Algorithmes. --- Algèbres commutatives --- Algebra - Data processing. --- ALGEBRA - data processing --- Algèbres commutatives --- Géométrie algébrique.

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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.

Algebra. --- Field theory (Physics). --- Algebraic geometry. --- Associative rings. --- Rings (Algebra). --- Category theory (Mathematics). --- Homological algebra. --- Differential equations. --- Partial differential equations. --- Field Theory and Polynomials. --- Algebraic Geometry. --- Associative Rings and Algebras. --- Category Theory, Homological Algebra. --- Ordinary Differential Equations. --- Partial Differential Equations. --- Gröbner bases. --- Gröbner basis theory --- Commutative algebra --- Partial differential equations --- 517.91 Differential equations --- Differential equations --- Homological algebra --- Algebra, Abstract --- Homology theory --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra) --- Algebraic geometry --- Geometry --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics --- Mathematics --- Mathematical analysis