This textbook about computer algebra gives an introduction to
this modern field of Mathematics. The contents of the first eight
chapters seem to be an indispensable foundation of computer algebra
and therefore form an introductory course. Chapters 9 to 12 build
the basis of a more advanced lecture on computer algebra and are
influenced by the author's own research interests.
Instead of describing algorithms—as in most other books—by so-called
pseudo code, all algorithms considered in this book are implemented
and presented as executable programs in the general-purpose computer
algebra system Mathematica. Nevertheless, lecturers are
absolutely not bound to this system, since the book’s definitions
and theorems are completely independent of a chosen computer algebra
system. Consequently, all computer algebra sessions can be
downloaded as worksheets in the three systems Mathematica, Maple
and Maxima from this internet page. Therefore every lecturer
can select his or her own favorite system. In particular, Maxima
is freely available and was used by the author when teaching the
same course several times in Africa at AIMS Cameroon.
By using real implementations instead of pseudo-code, the considered
algorithms are immediately applicable and verifiable. Knowledge from
higher algebra is not required, nevertheless all proofs are given.
Since the book is rather elementary and contains a detailed index,
it can also be used as a reference manual on algorithms of computer
The author lectured and did research in the field of computer
algebra in the last three decades and was the chairman of the
Computeralgebra from 2002-2011.
All sessions as zip-archivs: Mathematica, Maple, Maxima. I am indebted to Dr.
Bertrand Teguia Tabuguia who created the Maple and Maxima
Table of Contents
Chapter 1: Introduction to Computer Algebra . Capabilities
of Computer Algebra Systems
Chapter 2: Programming in Computer Algebra Systems .
Internal Representation of Expressions . Pattern Matching . Control
Structures . Recursion and Iteration . Remember Programming .
Divide-and-Conquer Programming . Programming through Pattern
Chapter 3: Number Systems and Integer Arithmetic .
Number Systems . Integer Arithmetic: Addition and
Multiplication . Integer Arithmetic: Division with Remainder .
The Extended Euclidean Algorithm . Unique Factorization . Rational
Chapter 4: Modular Arithmetic . Residue Class Rings .
Modulare Square Roots . Chinese Remainder Theorem . Fermat’s Little
Theorem . Modular Logarithms . Pseudoprimes
Chapter 5: Coding Theory and Cryptography . Basic
Concepts of Coding Theory . Prefix Codes . Check Digit Systems .
Error Correcting Codes . Asymmetric Ciphers
Chapter 6: Polynomial Arithmetic . Polynomial Rings .
Multiplication: The Karatsuba Algorithm . Fast Multiplication with
FFT . Division with Remainder . Polynomial Interpolation . The
Extended Euclidean Algorithm . Unique Factorization . Squarefree
Factorization . Rational Functions
Chapter 7: Algebraic Numbers . Polynomial Quotient
Rings . Chinese Remainder Theorem . Algebraic Numbers . Finite
Fields . Resultants . Polynomial Systems of Equations
Chapter 8: Factorization in Polynomial Rings .
Preliminary Considerations . Efficient Factorization in Zp[x]
. Squarefree Factorization of Polynomials over Finite Fields .
Efficient Factorization in Q[x] . Hensel Lifting .
Chapter 9: Simplification and Normal Forms . Normal
Forms and Canonical Forms . Normal Forms and Canonical Forms for
Polynomials . Normal Forms for Rational Functions . Normal Forms for
Chapter 10: Power Series . Formal Power Series .
Taylor Polynomials . Computation of Formal Power Series . Holonomic
Differential Equations . Holonomic Recurrence Equations .
Hypergeometric Functions . Efficient Computation of Taylor
Polynomials of Holonomic Functions . Algebraic Functions . Implicit
Chapter 11: Algorithmic Summation . Definite Summation
. Difference Calculus . Indefinite Summation . Indefinite Summation
of Hypergeometric Terms . Definite Summation of Hypergeometric Terms
Chapter 12: Algorithmic Integration . The Bernoulli
Algorithm for Rational Functions . Algebraic Prerequisites .
Rational Part . Logarithmic Case
References . List of Symbols . Mathematica List of Keywords . Index
July 05, 2021