Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

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Computability and Unsolvability.

A classical mathematical proof is constructed using pencil and paper. However, there are many ways in which computers may be used in a mathematical proof. But ‘proof by computer’, or even the use of computers in the course of a proof, is not so readily accepted (the December 2008 issue of the Notices of the American Mathematical Society is devoted to formal proofs by computer).In the following we introduce verification methods and discuss how they can assist in achieving a mathematically rigorous result. In particular we emphasize how floating-point arithmetic is used.

Verification methods: Rigorous results using floating-point arithmetic

Liouvillian functions are functions that are built up from rational functions using exponentiation, integration, and algebraic functions. We show that if a system of differential equations has a generic solution that satisfies a liouvillian relation, that is, there is a liouvillian function of several variables vanishing on the curve defined by this solution, then the system has a liouvillian first integral, that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set. We can refine this result in special cases to show that the first integral must be of a very special form

/pdf/liouvillian-first-integrals-of-differential-equations-chshub6qeh.pdf

Liouvillian first integrals of differential equations

Results which allow either the computatmn of symbolic solutions to first-order-linear difference equauons or the determination that solutions of a certain form do not exist are presented. Starting with a field of constants, larger fields may be constructed by the formal adjunctlon of symbols whtch behave hke solutions to first-order-linear equations (with a few restrictions) It IS in these extension fields that the difference equations may be posed and m which the solutions are requested. The principal apphcatmn of these results is In finding formulas for a broad class of finite sums or In showing the nonexistence of such formulas.

Summation in Finite Terms

This paper deals with the problem of telling whether a given elementary function, in the sense of analysis, has an elementary indefinite integral. In ?1 of this work, we give a precise definition of the elementary functions and develop the theory of integration of functions of a single varia' Z. By using functions of a complex, rather than a real variable, we can limit ourselves to exponentiation, taking logs, and algebraic operations in defining the elementary functions, since sin, tan- 1, etc., can be expressed in terms of these three. Following Ostrowski [9], we use the concept of a differential field. We strengthen the classical Liouville theorem and derive a number of consequences. ?2 uses the terminology of mathematical logic to discuss formulations of the problem of integration in finite terms. ?3 (the major part of this paper) uses the previously developed theory to give an algorithm for determining the elementary integrability of those elementary functions which can be built up (roughly speaking) using only the rational operations, exponentiation and taking logarithms; however, if these exponentiations and logarithms can be replaced by adjoining constants and performing algebraic operations, the algorithm, as it is presented here, cannot be applied. The man who established integration in finite terms as a mathematical discipline was Joseph Liouville (1809-1882), whose work on this subject appeared in the years 1833-1841. The Russian mathematician D. D. Mordoukhay-Boltovskoy (1876-1952) wrote much on this and related matters. The present writer received his introduction to this subject through the book [10] by the American J. F. Ritt

/pdf/the-problem-of-integration-in-finite-terms-lq3xpw8i2c.pdf

The problem of integration in finite terms

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The problem of integration in finite terms