Computers in Mathematics

V. Chudnovsky, Richard D. Jenks

• Publisher: CRC Press
• Year: 2020
• Print ISBN: 9781138413429
• eBook ISBN: 9781000153705
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

Talks from the International Conference on Computers and Mathematics held July 29-Aug. 1, 1986, Stanford U. Some are focused on the past and future roles of computers as a research tool in such areas as number theory, analysis, special functions, combinatorics, algebraic geometry, topology, physics,

Computer Algebra in the Service of Mathematical Physics and Number Theory

DAVID V. CHUDNOVSKY GREGORY V. CHUDNOVSKY Columbia University, New York, New York

1 Introduction.

Methods of computer algebra become more familiar to a wide audience of theoretical mathematicians and physicists. The environment of computer algebra system leads to a greater acceptance of computer instruments in the mathematical research. Methods of symbolic manipulation provided by computer algebra systems in combination with high-power number crunching abilities of traditional hardware and software open the way to truly large scale computations often needed by mathematicians and physicists.

These possibilities of better servicing the everyday needs of the researcher emphasize as a priority the development of new algorithms and methods of efficient programming solution of basic computational tasks of theoretical and applied mathematics. Considerable progress in this area is associated with classes of problems, when explicit complexity description exists, like the study of algebraic complexities of the basic algebraic tasks: polynomial operations, matrix multiplication, solution of problems of linear algebra and linear programming. In some basic number-theoretic problems important progress in the construction of new algorithms is associated with the use of algebraic geometry (primality testing and factorization). Many of new algorithms and methods of number-theoretic computations and computer algebra programming are discussed in detail in the talks presented at this conference. We decided to focus on a few important computational problems, whose implementation often requires considerable number-crunching effort. At the cornerstone of these problems lies the task of evaluation and tabulation of values of transcendental and algebraic functions—a task of classical numerical analysis. Our specific interest in these problems arose from our work with computer algebra systems, whose environment gives a use a wonderful capability of multiple precision computations. This capability of computing with full precision (for big integers), and with very high precision to satisfy the curiosity of a diophantine geometer, led to the responsibility of efficient programming. We describe new low complexity (both operational and logical) methods and algorithms for computations of solutions of differential equations and their efficient evaluation, and of solution of algebraic equations. These computations are often basic in applied problems, but even more so in a variety of problems of number theory and algebraic geometry (topology). This includes computations (in positive its multidimensional generalization to systems of equations, is widely recognized as one of the most cumbersome issues in c...

Table of contents

• Cover
• Half Title
• Pure and Applied Mathematics
• Lecture Notes
• Title Page
• Copyright Page
• Preface
• Table of Contents
• Contributors
• Keynote Remarks at the Conference on Computers and Mathematics
• SCRATCHPAD Explorations for Elliptic Theta Functions
• Integration and Computers
• Some Thoughts on Proof Discovery
• Computer Algebra in the Service of Mathematical Physics and Number Theory
• Impact of Linear Programming on Computer
• Uses of and Limitations of Computers in Number Theory
• Strip Mining in the Abandoned Orefields of Nineteenth Century Mathematics
• Polynomial Factorization 1982–1986
• Factorization Then and Now
• Computer Animation in Mathematics, Science, and Art
• Physicists and Computer Algebra
• Symbolic Computation: The Early Days (1950-1971)
• What Computer Algebra Systems Can and Cannot Do
• Solution of Equations I: Polynomial Ideals and Gröbner Bases
• Index