# Computer Algebra Handbook

## Foundations · Applications · Systems

• Johannes Grabmeier
• Erich Kaltofen
• Volker Weispfenning
Book

1. Front Matter
Pages i-xx
2. Wolfram Koepf, Erich Kaltofen, Volker Weispfenning, Johannes Grabmeier, B. Heinrich Matzat
Pages 1-9
3. Rüdiger Loos, M. Giusti, L. M. Pardo, Gerhard Niklasch, Ralf Dentzer, Susan Landau et al.
Pages 11-162
4. Friedrich W. Hehl, Jochem Fleischer, Matthias Steinhauser, Georg Weiglein, Jos Vermaseren, Christian Heinicke et al.
Pages 163-260
5. Michael Dewar, Johannes Grabmeier, Stephen Watt, Bernhard Kutzler, Mascyma Technical Staff, John Cannon et al.
Pages 261-483
6. Johannes Grabmeier, Erich Kaltofen, Volker Weispfenning
Pages 485-492
7. Back Matter
Pages 493-637

### Introduction

Two ideas lie gleaming on the jeweler's velvet. The first is the calculus, the sec­ ond, the algorithm. The calculus and the rich body of mathematical analysis to which it gave rise made modern science possible; but it has been the algorithm that has made possible the modern world. -David Berlinski, The Advent of the Algorithm First there was the concept of integers, then there were symbols for integers: I, II, III, 1111, fttt (what might be called a sticks and stones representation); I, II, III, IV, V (Roman numerals); 1, 2, 3, 4, 5 (Arabic numerals), etc. Then there were other concepts with symbols for them and algorithms (sometimes) for ma­ nipulating the new symbols. Then came collections of mathematical knowledge (tables of mathematical computations, theorems of general results). Soon after algorithms came devices that provided assistancefor carryingout computations. Then mathematical knowledge was organized and structured into several related concepts (and symbols): logic, algebra, analysis, topology, algebraic geometry, number theory, combinatorics, etc. This organization and abstraction lead to new algorithms and new fields like universal algebra. But always our symbol systems reflected and influenced our thinking, our concepts, and our algorithms.

### Keywords

Hardware algebra algorithms coding theory complexity complexity theory computer computer algebra computer algebra system computer science cryptography geometry linear algebra logic number theory

### Editors and affiliations

• Johannes Grabmeier
• 1
• Erich Kaltofen
• 2
• Volker Weispfenning
• 3
1. 1.University of Applied SciencesDeggendorfGermany
2. 2.Department of MathematicsNorth carolina State UniversityRaleighUSA
3. 3.Department of MathematicsUniversity of PassauPassauGermany

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-642-55826-9
• Copyright Information Springer-Verlag Berlin Heidelberg 2003
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-642-62988-4
• Online ISBN 978-3-642-55826-9
• Buy this book on publisher's site