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This monograph strives to introduce a solid foundation on the usage of Gröbner 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 Gröbner 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 Gröbner bases in determining and recognizing many more structural properties of algebras, such as the Gelfand-Kirillov dimension, Noetherianity, (semi-)primeness, PI-property, finitenes

Gröbner bases. --- Rings (Algebra) --- Gröbner basis theory --- Commutative algebra --- Algebraic rings --- Ring theory --- Algebraic fields --- Gröbner, Wolfgang, --- Gröbner, W.

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The second volume of this comprehensive treatise focusses on Buchberger theory and its application to the algorithmic view of commutative algebra. In distinction to other works, the presentation here is based on the intrinsic linear algebra structure of Groebner bases, and thus elementary considerations lead easily to the state-of-the-art in issues of implementation. The same language describes the applications of Groebner technology to the central problems of commutative algebra. The book can be also used as a reference on elementary ideal theory and a source for the state-of-the-art in its algorithmization. Aiming to provide a complete survey on Groebner bases and their applications, the author also includes advanced aspects of Buchberger theory, such as the complexity of the algorithm, Galligo's theorem, the optimality of degrevlex, the Gianni-Kalkbrener theorem, the FGLM algorithm, and so on. Thus it will be essential for all workers in commutative algebra, computational algebra and algebraic geometry.

Equations --- Polynomials. --- Iterative methods (Mathematics) --- Iteration (Mathematics) --- Numerical analysis --- Algebra --- Numerical solutions. --- Graphic methods --- Gröbner bases. --- Gröbner basis theory --- Commutative algebra

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This volume contains survey and original articles presenting the state of the art on the application of Gröbner bases in control theory and signal processing. The contributions are based on talks delivered at the Special Semester on Gröbner Bases and Related Methods at the Johann Radon Institute of Computational and Applied Mathematics (RICAM), Linz, Austria, in May 2006.

Gröbner bases. --- Control theory. --- Signal processing. --- Processing, Signal --- Information measurement --- Signal theory (Telecommunication) --- Dynamics --- Machine theory --- Gröbner basis theory --- Commutative algebra --- Algebras. --- base. --- control theory. --- signal theory. --- Grobner bases.

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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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Computational Commutative Algebra 2 is the natural continuation of Computational Commutative Algebra 1 with some twists, starting with the differently coloured cover graphics. The first volume had 3 chapters, 20 sections, 44 tutorials, and some amusing quotes. Since bigger is better, this book contains 3 chapters filling almost twice as many pages, 23 sections (some as big as a whole chapter), and 55 tutorials (some as big as a whole section). The number of jokes and quotes has increased exponentially due to the little-known fact that a good mathematical joke is better than a dozen mediocre papers. The main part of this book is a breathtaking passeggiata through the computational domains of graded rings and modules and their Hilbert functions. Besides Gröbner bases, we encounter Hilbert bases, border bases, SAGBI bases, and even SuperG bases. The tutorials traverse areas ranging from algebraic geometry and combinatorics to photogrammetry, magic squares, coding theory, statistics, and automatic theorem proving. Whereas in the first volume gardening and chess playing were not treated, in this volume they are. This is a book for learning, teaching, reading, and most of all, enjoying the topic at hand. The theories it describes can be applied to anything from children's toys to oil production. If you buy it, probably one spot on your desk will be lost forever!

Gröbner bases --- Commutative algebra --- Grobner, Bases de --- Algebre commutative --- Data processing --- Informatique --- Commutative algebra -- Data processing. --- Gröbner bases. --- Gröbner basis theory --- Mathematics. --- Computer science --- Algebra. --- Algebraic geometry. --- Computer mathematics. --- Algorithms. --- Computational Mathematics and Numerical Analysis. --- Symbolic and Algebraic Manipulation. --- Algebraic Geometry. --- Data processing. --- Algebra --- Geometry, algebraic. --- Algebraic geometry --- Geometry --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Algorism --- Arithmetic --- Mathematics --- Mathematical analysis --- Foundations --- Combinatorial analysis. --- QA 150-272 Algebra. --- Computer science—Mathematics. --- Group theory. --- Group Theory and Generalizations. --- Groups, Theory of --- Substitutions (Mathematics) --- Commutative algebra - Data processing --- Algebre commutative - Informatique