| Preface | |
| Introduction to Computer Algebra | |
| Symbolic versus Numeric Computation | p. 2 |
| A Brief Historical Sketch | p. 4 |
| An Example of a Computer Algebra System: MAPLE | p. 11 |
| Algebra of Polynomials, Rational Functions, and Power Series | |
| Rings and Fields | p. 23 |
| Divisibility and Factorization in Integral Domains | p. 26 |
| The Euclidean Algorithm | p. 32 |
| Univariate Polynomial Domains | p. 38 |
| Multivariate Polynomial Domains | p. 46 |
| The Primitive Euclidean Algorithm | p. 52 |
| Quotient Fields and Rational Functions | p. 60 |
| Power Series and Extended Power Series | p. 63 |
| Relationships among Domains | p. 70 |
| Normal Forms and Algebraic Representations | |
| Levels of Abstraction | p. 79 |
| Normal Form and Canonical Form | p. 80 |
| Normal Forms for Polynomials | p. 84 |
| Normal Forms for Rational Functions and Power Series | p. 88 |
| Data Structures for Multiprecision Integers and Rational Numbers | p. 93 |
| Data Structures for Polynomials, Rational Functions, and Power Series | p. 96 |
| Arithmetic of Polynomials, Rational Functions, and Power Series | |
| Basic Arithmetic Algorithms | p. 112 |
| Fast Arithmetic Algorithms: Karatsuba's Algorithm | p. 118 |
| Modular Representations | p. 120 |
| The Fast Fourier Transform | p. 123 |
| The Inverse Fourier Transform | p. 128 |
| Fast Polynomial Multiplication | p. 132 |
| Computing Primitive N-th Roots of Unity | p. 133 |
| Newton's Iteration for Power Series Division | p. 136 |
| Homomorphisms and Chinese Remainder Algorithms | |
| Intermediate Expression Swell: An Example | p. 151 |
| Ring Morphisms | p. 153 |
| Characterization of Morphisms | p. 160 |
| Homomorphic Images | p. 167 |
| The Integer Chinese Remainder Algorithm | p. 174 |
| The Polynomial Interpolation Algorithm | p. 183 |
| Further Discussion of the Two Algorithms | p. 189 |
| Newton's Iteration and the Hensel Construction | |
| P-adic and Ideal-adic Representations | p. 205 |
| Newton's Iteration for F(u)=0 | p. 214 |
| Hensel's Lemma | p. 226 |
| The Univariate Hensel Lifting Algorithm | p. 232 |
| Special Techniques for the Non-monic Case | p. 240 |
| The Multivariate Generalization of Hensel's Lemma | p. 250 |
| The Multivariate Hensel Lifting Algorithm | p. 260 |
| Polynomial GCD Computation | |
| Polynomial Remainder Sequences | p. 280 |
| The Sylvester Matrix and Subresultants | p. 285 |
| The Modular GCD Algorithm | p. 300 |
| The Sparse Modular GCD Algorithm | p. 311 |
| GCD's using Hensel Lifting: The EZ-GCD Algorithm | p. 314 |
| A Heuristic Polynomial GCD Algorithm | p. 320 |
| Polynomial Factorization | |
| Square-Free Factorization | p. 337 |
| Square-Free Factorization Over Finite Fields | p. 343 |
| Berlekamp's Factorization Algorithm | p. 347 |
| The Big Prime Berlekamp Algorithm | p. 359 |
| Distinct Degree Factorization | p. 368 |
| Factoring Polynomials over the Rationals | p. 374 |
| Factoring Polynomials over Algebraic Number Fields | p. 378 |
| Solving Systems of Equations | |
| Linear Equations and Gaussian Elimination | p. 390 |
| Fraction-Free Gaussian Elimination | p. 393 |
| Alternative Methods for Solving Linear Equations | p. 399 |
| Nonlinear Equations and Resultants | p. 405 |
| Grobner Bases for Polynomial Ideals | |
| Term Orderings and Reduction | p. 431 |
| Grobner Bases and Buchberger's Algorithm | p. 439 |
| Improving Buchberger's Algorithm | p. 447 |
| Applications of Grobner Bases | p. 451 |
| Additional Applications | p. 462 |
| Integration of Rational Functions | |
| Basic Concepts of Differential Algebra | p. 474 |
| Rational Part of the Integral: Hermite's Method | p. 482 |
| Rational Part of the Integral: Horowitz' Method | p. 488 |
| Logarithmic Part of the Integral | p. 492 |
| The Risch Integration Algorithm | |
| Elementary Functions | p. 512 |
| Differentiation of Elementary Functions | p. 519 |
| Liouville's Principle | p. 523 |
| The Risch Algorithm for Transcendental Elementary Functions | p. 529 |
| The Risch Algorithm for Logarithmic Extensions | p. 530 |
| The Risch Algorithm for Exponential Extensions | p. 547 |
| Integration of Algebraic Functions | p. 561 |
| Notation | p. 575 |
| Index | p. 577 |
| Table of Contents provided by Blackwell. All Rights Reserved. |