Introduction; 1. Cyclohexane, cryptography, codes, and computer algebra; 2. Fundamental algorithms; 3. The Euclidean algorithm; 4. Applications of the Euclidean algorithm; 5. Interpolation and modular algorithms; 6. The resultant and gcd computation; 7. Application: decoding BCH codes; 8. Fast multiplication; 9. Newton iteration; 10. Fast polynomial evaluation and interpolation; 11. Fast Euclidean algorithm; 12. Fast linear algebra; 13. Fourier transforms, signal processing, and image compression; 14. Factoring polynomials over finite fields; 15. Hensel lifting and factoring polynomials; 16. Short vectors in lattices; 17. Applications of basis reduction; 18. Primality testing; 19. Factoring integers; 20. Application: public key cryptography; 21. Gröbner bases; 22. Symbolic integration; 23. Symbolic summation; 24. Applications; 25. Fundamental concepts; References.
Graduate text on computer algebra for computer science and mathematics students.
' ... a most fascinating and readable survey of methods that are used by modern Computer Algebra Systems (CAS's). This book is a delight; I heartly recommend it (a) to anybody who has any interest at all in how their CAS works and (b) to anybody who enjoys seeing elegant and rigorous mathematics applied to practical problems.' Alistair Fitt, University of Southampton 'Wow! What a beautifully produced book, and what a wealth of information! Also written in perfect idiomatic English by two German authors' Don Knuth ' ... a pleasure to read. I predict it will be a major success.' Steve Cook 'It's really an impressive work. I'm sure it will become a reference for computer algebra algorithms, and stay it for a long time.' Paul Zimmermann 'This book is a delight; I heartily recommend it to anybody who has interest at all in how their CAS works and anybody who enjoys seeing elegant and rigorous mathematics applied to practical problems.' Alistair Fitt 'On each page I can feel thorough understanding and love for the subject and uncompromising scholarship in presenting its story. A masterpiece.' Erich Kaltofen ' ... a wonderful book! I am working my way through it (for what will probably be the first of many times), and am very pleased with it ... aesthetically beautiful, with the color plots and multitudes of epigraphs for each chapter.' Todd Moon ' ... it looks stunning! ...' Hendrik Lenstra ' ... a most successful balance of intuition, rigorous mathematics, interesting and beautiful applications and completeness. The extensive collection of exercises makes it into an ideal textbook for the use in a graduate course.' Ton Levelt 'The style is very clear. The arguments are very precise. In my opinion, the level of presentation is convenient for graduate students, some parts are quite readable for good undergraduate students. The typography and the quality of illustration of this Cambridge book are splendid. To conclude, I find the quality of this book really exceptional, this is certainly an excellent source on the state of the art in computer algebra.' Maurice Mignottte, Zbl. MATH '... a welcome survey of a fast-gorwing and toical area of modern mathematics and computer science.' European Maths Society Journal
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