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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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The theory of Gröbner bases, invented by Bruno Buchberger, is a general method by which many fundamental problems in various branches of mathematics and engineering can be solved by structurally simple algorithms. The method is now available in all major mathematical software systems. This book provides a short and easy-to-read account of the theory of Gröbner bases and its applications. It is in two parts, the first consisting of tutorial lectures, beginning with a general introduction. The subject is then developed in a further twelve tutorials, written by leading experts, on the application of Gröbner bases in various fields of mathematics. In the second part are seventeen original research papers on Gröbner bases. An appendix contains the English translations of the original German papers of Bruno Buchberger in which Gröbner bases were introduced.

Gröbner bases. --- Coding theory --- Gröbner basis theory --- Commutative algebra --- Geometry, Algebraic --- Géométrie algébrique. --- Algorithms --- Algorithmes. --- Anneaux de polynômes. --- Polynomial rings. --- Algèbres commutatives. --- Polynômes --- Polynomials --- 512.56 --- 512.56 Lattices, including Boolean rings and algebras --- Lattices, including Boolean rings and algebras --- Grobner bases. --- Géométrie algébrique --- Algorithmes --- Anneaux de polynômes. --- Algèbres commutatives. --- Polynômes