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2021 | Book

Introduction to Computer Algebra Read chapter Read first chapter

Computer Algebra

An Algorithm-Oriented Introduction

Author: Prof. Dr. Wolfram Koepf

Publisher: Springer International Publishing

Book Series : Springer Undergraduate Texts in Mathematics and Technology

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This textbook offers an algorithmic introduction to the field of computer algebra. A leading expert in the field, the author guides readers through numerous hands-on tutorials designed to build practical skills and algorithmic thinking. This implementation-oriented approach equips readers with versatile tools that can be used to enhance studies in mathematical theory, applications, or teaching. Presented using Mathematica code, the book is fully supported by downloadable sessions in Mathematica, Maple, and Maxima.

Opening with an introduction to computer algebra systems and the basics of programming mathematical algorithms, the book goes on to explore integer arithmetic. A chapter on modular arithmetic completes the number-theoretic foundations, which are then applied to coding theory and cryptography. From here, the focus shifts to polynomial arithmetic and algebraic numbers, with modern algorithms allowing the efficient factorization of polynomials. The final chapters offer extensions into more advanced topics: simplification and normal forms, power series, summation formulas, and integration.

Computer Algebra is an indispensable resource for mathematics and computer science students new to the field. Numerous examples illustrate algorithms and their implementation throughout, with online support materials to encourage hands-on exploration. Prerequisites are minimal, with only a knowledge of calculus and linear algebra assumed. In addition to classroom use, the elementary approach and detailed index make this book an ideal reference for algorithms in computer algebra.

Table of Contents

Frontmatter
Chapter 1. Introduction to Computer Algebra
Abstract
Before we discuss mathematical algorithms and their programming, we want to show the capabilities of a general computer algebra system such as Mathematica (The same questions can be also treated with the systems Maple and Maxima, and the corresponding worksheets can be downloaded from www.​computer-algebra.​org).
Wolfram Koepf
Chapter 2. Programming in Computer Algebra Systems
Abstract
In Mathematica, the internal representation of every expression is a list structure of the form Head [Argument1,Argument2,., Argument n] with finitely many arguments. The following table gives some simple examples for internal representations.
Wolfram Koepf
Chapter 3. Number Systems and Integer Arithmetic
Abstract
In this chapter we consider the arithmetics of arbitrary large integers. Integer arithmetic is the backbone of every computer algebra system, hence we would like to discuss this topic in some detail. In order to be able to compute with integers, they must be suitably represented. In the history of Mathematics, several representations of integers have been introduced, two of which are still in use: the roman and the arabic number systems.
Wolfram Koepf
Chapter 4. Modular Arithmetic
Abstract
The integers with their operations addition and multiplication \((\mathbb{Z}, +, \cdot)\) form a commutative ring with unity 1. Now, we would like to translate this algebraic structure to finite subsets of \(\mathbb{Z}\). This is done by identification of certain elements in \(\mathbb{Z}\) that lie in common arithmetic progressions.
Wolfram Koepf
Chapter 5. Coding Theory and Cryptography
Abstract
Coding Theory deals with the secure transfer of messages. We assume that the sender (source) of the message sends the message via a communication channel to the recipient. The type of coding depends on the practical situation, hence the communication channel might also be disturbed.
Wolfram Koepf
Chapter 6. Polynomial Arithmetic
Abstract
In the sequel (if not stated otherwise), we assume that R is a zero-divisor-free ring with unity 1. Such a ring is called an integral domain.
Wolfram Koepf
Chapter 7. Algebraic Numbers
Abstract
In the same way as we defined the computation modulo an integer \(p \in \mathbb{Z}\) and the resulting remainder class rings in Chapter 4, the computation modulo a polynomial \(p(x) \in {\mathbb{K}}[x]\) can be defined. Again, this results in remainder class rings, which are now called polynomial quotient rings.
Wolfram Koepf
Chapter 8. Factorization in Polynomial Rings
Abstract
The aim of this chapter is the development of efficient algorithms for factorization in ℚ[x]. For this purpose, efficient factorization algorithms for ℤp[x] are used which, of course, are interesting by themselves.
Wolfram Koepf
Chapter 9. Simplification and Normal Forms
Abstract
If we call a simplification command like Simplify, FullSimplify, Expand or Together in Mathematica, then algebraic expressions are replaced by (hopefully) mathematically equivalent ones. Therefore the general question arises under which circumstances such transformations are possible and which types of simplifications can be executed.
Wolfram Koepf
Chapter 10. Power Series
Abstract
If R is an integral domain, then the same is true for the set R[x] of polynomials. Addition and multiplication in R[x] were defined in Section 6.1. Analogously one can define the integral domain of formal power series R[[x]].
Wolfram Koepf
Chapter 11. Algorithmic Summation
Abstract
Since a rather simple (although not very efficient) algorithm exists for them we would first like to discuss bivariate sums of a specific form.
Wolfram Koepf
Chapter 12. Algorithmic Integration
Abstract
In the previous chapter, we studied, among other things, the algorithmic computation of antidifferences in the special case where these antidifferences are hypergeometric terms. In this chapter, we would like to consider algorithmic integration.
Wolfram Koepf
Backmatter
Metadata
Title
Computer Algebra
Author
Prof. Dr. Wolfram Koepf
Copyright Year
2021
Electronic ISBN
978-3-030-78017-3
Print ISBN
978-3-030-78016-6
DOI
https://doi.org/10.1007/978-3-030-78017-3

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