Synopsis
1 Development, Characterization, Prospects.- 1.1 Historical Remarks.- 1.2 General Characterization.- 1.3 Impact on Education.- 1.4 Impact on Research.- 1.5 Computer Algebra - Today and Tomorrow.- 1.5.1 Today.- 1.5.2 Outlook.- 2 Topics of Computer Algebra.- 2.1 Exact Arithmetic.- 2.1.1 Long Integer Arithmetic.- 2.1.2 Arithmetic with Polynomials, Rational Functions and Power Series.- 2.1.3 Euchd's Algorithm and Continued Fractions.- 2.1.4 Modular Arithmetic and the Chinese Remainder Theorem.- 2.1.5 Computations with Algebraic Numbers.- 2.1.6 Real Algebraic Numbers.- 2.1.7 p-adic Numbers and Approximations.- 2.1.8 Finite Fields.- 2.2 Algorithms for Polynomials and Power Series.- 2.2.1 The Division Algorithm.- 2.2.2 Factorization of Polynomials.- 2.2.3 Absolute Factorization of Polynomials.- 2.2.4 Polynomial Decomposition.- 2.2.5 Gröbner Bases.- 2.2.6 Standard Bases.- 2.2.7 Characteristic Sets.- 2.2.8 Algorithmic Invariant Theory.- 2.3 Linear Algebra.- 2.3.1 Linear Systems.- 2.3.2 Algorithms for Matrix CanonicalForms.- 2.4 Constructive Methods of Number Theory.- 2.4.1 Primality Tests.- 2.4.2 Integer Factorization.- 2.4.3 Algebraic Number Fields and AlgebraicFunction Fields.- 2.4.4 Galois Groups.- 2.4.5 Rational Points on Elhptic Curves.- 2.4.6 Geometry of Numbers.- 2.5 Algorithms of Commutative Algebra and Algebraic Geometry.- 2.5.1 Algorithms for Polynomial Ideals and Their Varieties.- 2.5.2 Singularities of Varieties.- 2.5.3 Real Algebraic Geometry.- 2.6 Algorithmic Aspects of the Theory of Algebras.- 2.6.1 Structure Constants.- 2.6.2 Generators and Relations, Swapping and G-Algebras.- 2.6.3 Monad Algebras, Path Algebras and Generalizations.- 2.6.4 Finite-Dimensional Lie Algebras.- 2.6.5 Non-commutative Gröbner Bases.- 2.6.6 Structural Issues and Classification.- 2.6.7 Identities.- 2.6.8 Computational Aspects in the Representation Theory of Quivers and Path Algebras.- 2.7 Computational Group Theory.- 2.7.1 A Crash Course in Group Theory.- 2.7.2 Describing Groups.- 2.7.3 A Brief History.- 2.7.4 Permutation Groups.- 2.7.5 Matrix Groups.- 2.7.6 Black Box Groups.- 2.7.7 Abelian Groups.- 2.7.8 Polycyclic Groups.- 2.7.9 Finitely Presented Groups.- 2.7.10 Group-Theoretic Software.- 2.7.11 Another Perspective.- 2.8 Algorithms of Representation Theory.- 2.8.1 Ordinary Representation Theory.- 2.8.2 Modular Representation Theory.- 2.8.3 Generic Character Tables.- 2.8.4 Summary of Systems.- 2.9 Algebraic Methods for Constructing Discrete Structures.- 2.10 Summation and Integration.- 2.10.1 Definite Summation and Hypergeometric Identities.- 2.10.2 Symbolic Integration.- 2.11 Symbohc Methods for DiflFerential Equations.- 2.11.1 Introduction.- 2.11.2 Differential Galois Theory.- 2.11.3 Lie Symmetries.- 2.11.4 Painlevé Theory.- 2.11.5 Completion.- 2.11.6 Differential Ideal Theory.- 2.11.7 Dynamical Systems.- 2.11.8 Numerical Analysis.- 2.12 Symbolic/Numeric Methods.- 2.12.1 Computer Analysis.- 2.12.2 Algorithms for Computing Validated Results.- 2.12.3 Hybrid Methods.- 2.13 Algebraic Complexity Theory.- 2.14 Coding Theory and Cryptography.- 2.14.1 Coding Theory.- 2.14.2 QuantumCoding Theory.- 2.14.3 Cryptography.- 2.15 Algorithmic Methods inUniversal Algebra and Logic.- 2.15.1 Term Rewriting Systems.- 2.15.2 Decision Procedures and Quantifier Ehmination Methods for AlgebraicTheories.- 2.16 Knowledge Representation and Abstract Data Types.- 2.16.1 Mathematical Knowledge Representation and Expert Systems.- 2.16.2 Abstract Data Types.- 2.17 On the Design of Computer Algebra Systems.- 2.17.1 Memory Management.- 2.17.2 Program Verification and Abstract Data Types.- 2.17.3 The Concept of Types.- 2.17.4 Genericity.- 2.17.5 Modularization.- 2.17.6 Parallel Implementation.- 2.17.7 Continuing Developmentof Computer Algebra Systems.- 2.18 Parahel Computer Algebra Systems.- 2.18.1 Parallel Architectures and Operating Systems Supports.- 2.18.2 Parallel Execution: Mapping and Scheduling.- 2.18.3 Parallelism Expression and Languages.- 2.19 Interfaces and S
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