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Some tapas of computer algebra. (English) Zbl 0924.13021
Algorithms and Computation in Mathematics. 4. Berlin: Springer. xiv, 352 p. (1999).

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The book grows out of a series of lectures of two mini courses held at the Research School of the Euler Institute for Discrete Mathematics and its Applications in Eindhoven in the years 1994 and 1995. The idea of the courses was to acquaint young mathematicians with algorithmic aspects of pure mathematics as well as to enable them to incorporate algorithms in their own research. The editors’ intention was to underline that the courses consists of appetizers (tapas) for various parts of algorithmic algebra covering such ‘spicy’ relations of Gröbner bases to integer programming and algebraic solutions of differential equations.
The book is different from a textbook about computer algebra in the following sense. It provides a short introduction to Gröbner bases (Chapter 1: Gröbner bases, an introduction (1-33) (by A. Cohen)) and some basic key results as the LLL algorithm (Chapter 3: Lattice reduction (66-77) (by F. Beukers) and factorization (Chapter 4: Factorization of polynomials (78-90) (by F. Beukers)).
One of the most important features of symbolic methods it the solution of polynomial equations in several variables. This is studied in chapter 2: Symbolic recipes for polynomial system solving (34-65) (by L. Gonzalez-Vega, F. Rouillier, M.-F. Roy) and chapter 6: Symbolic recipes for real solutions (121-167) (by L. Gonzalez-Vega, F. Rouillier, M.-F. Roy, G. Trujillo).
Gröbner bases theory is the foundation of the applications concerning integer programming (Chapter 7: Gröbner bases and integer programming (168-183) (by G. M. Ziegler)) and coding theory (Chapter 10: Gröbner bases for codes (237-259) (by M. de Boer and R. Pellikaan) and Chapter 11: Gröbner bases for decoding (260-275) (by M. de Boer and R. Pellikaan)).
Chapter 5: Computations in associative and Lie algebras (91-120) (by G. Ivanyos and L. Rónyai) is concerned with algorithmic aspects of non-commutative algebra. Symbolic methods related to group theoretic methods, respectively to differential equations are discussed in chapter 8: Working with finite groups (184-207) (by H. Cuypers, L. H. Soicher and H. Sterk) respectively in chapter 9: Symbolic analysis of differential equations (208-236) (by M. van der Put).
These eleven lectures are completed by seven ‘projects’ for those wanting to acquaint themselves somewhat further with part of the material of the lectures. It is a coherent body of exercises around a theme which could serve as a practical session to the lectures. The projects are 1. Automatic geometry theorem proving (276-296) (by T. Recio, H. Sterk and M. Pilar Vélez), 2. The Birkhoff interpolation problems (297-304) (by M.-J. Gonzales-Lopez and L. Gonzalez-Vega), 3. The inverse kinematic problem in robotics (305-310) (by M.-J. Gonzalez-Lopez and L. Gonzalez-Vega), 4. Quaternion algebras (311-314) (by G. Ivanyos and L. Rónyai), 5. Explorations with the icosahedral group (315-322) (by A. M. Cohen, H. Cuypers and R. Riebeck), 6. The small Matthieu groups (323-337) (by H. Cuypers, L. H. Soicher and H. Sterk), and 7. The Golay codes (338-347) (by M. de Boer and R. Pellikaan).
The editors addressed the book to university teachers for composing a course in mathematical aspects of computer algebra as well as to advanced undergraduate students interested in algorithms in algebra. The material enlarges the known applications of symbolic methods by further interesting applications. A large number of references – separately for each chapter and each project – provide the reader with the knowledge of the present day research on the subject of the book.
Most of the chapters and projects will be reviewed individually.
Reviewer: P.Schenzel (Halle)

MSC:
13Pxx Computational aspects and applications of commutative rings
13-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra
14Pxx Real algebraic and real-analytic geometry
14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry
68W10 Parallel algorithms in computer science
94-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to information and communication theory
Software:
nauty
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