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This volume contains survey articles and original research papers, presenting the state of the art on applying the symbolic approach of Gröbner bases and related methods to differential and difference equations. The contributions are based on talks delivered at the Special Semester on Gröbner Bases and Related Methods hosted by the Johann Radon Institute of Computational and Applied Mathematics, Linz, Austria, in May 2006.

Gröbner bases --- Differential equations --- 517.91 Differential equations --- Gröbner basis theory --- Commutative algebra --- Kongress. --- Linz <2006> --- Algebras. --- base. --- ideal. --- polynomial. --- sybolic computation.

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Ordered algebraic structures --- Gröbner bases --- Convex polytopes --- 512.62 --- Grobner bases --- Polytopes --- Gröbner basis theory --- Commutative algebra --- Fields. Polynomials --- Convex polytopes. --- Gröbner bases. --- 512.62 Fields. Polynomials --- Gröbner bases

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This volume consists of research papers and expository survey articles presented by the invited speakers of the conference on "Harmony of Gröbner Bases and the Modern Industrial Society". Topics include computational commutative algebra, algebraic statistics, algorithms of D-modules and combinatorics. This volume also provides current trends on Gröbner bases and will stimulate further development of many research areas surrounding Gröbner bases.

Gröbner bases --- Gröbner basis theory --- Commutative algebra --- Developed countries --- Advanced countries --- Advanced nations --- Developed nations --- Economically advanced countries --- Economically advanced nations --- First World --- Industrial countries --- Industrial nations --- Industrial societies --- Industrialized countries --- Industrialized nations --- Western countries

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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.

Gröbner bases. --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Gröbner basis theory --- Statistics. --- Statistical Theory and Methods. --- Statistics and Computing/Statistics Programs. --- Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences. --- Commutative algebra --- Mathematical statistics. --- Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Econometrics --- Statistical inference --- Statistics, Mathematical --- Statistics --- Probabilities --- Sampling (Statistics) --- Statistics .

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This book provides an easy-to-read account of the theory of Gröbner bases and applications. It is in 2 parts, the first consists of tutorial lectures, and the second, 17 original research papers on Gröbner bases.

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