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Computational Excursions in Analysis and Number Theory

  • Book
  • © 2002

Overview

Authors:
  1. Peter Borwein

    1. Department of Mathematics and Statistics, Simon Fraser University, Burnaby, Canada

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Part of the book series: CMS Books in Mathematics (CMSBM)

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Table of contents (16 chapters)

  1. Front Matter

    Pages i-x
    Download chapter PDF

  2. Introduction

    • Peter Borwein
    Pages 1-9

  3. LLL and PSLQ

    • Peter Borwein
    Pages 11-13

  4. Pisot and Salem Numbers

    • Peter Borwein
    Pages 15-26

  5. Rudin—Shapiro Polynomials

    • Peter Borwein
    Pages 27-35

  6. Fekete Polynomials

    • Peter Borwein
    Pages 37-42

  7. Products of Cyclotomic Polynomials

    • Peter Borwein
    Pages 43-52

  8. Location of Zeros

    • Peter Borwein
    Pages 53-58

  9. Maximal Vanishing

    • Peter Borwein
    Pages 59-66

  10. Diophantine Approximation of Zeros

    • Peter Borwein
    Pages 67-74

  11. The Integer Chebyshev Problem

    • Peter Borwein
    Pages 75-84

  12. The Prouhet—Tarry—Escott Problem

    • Peter Borwein
    Pages 85-95

  13. The Easier Waring Problem

    • Peter Borwein
    Pages 97-101

  14. The Erdős—Szekeres Problem

    • Peter Borwein
    Pages 103-107

  15. Barker Polynomials and Golay Pairs

    • Peter Borwein
    Pages 109-119

  16. The Littlewood Problem

    • Peter Borwein
    Pages 121-132

  17. Spectra

    • Peter Borwein
    Pages 133-139

  18. Back Matter

    Pages 141-220
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Keywords

About this book

This book is designed for a topics course in computational number theory. It is based around a number of difficult old problems that live at the interface of analysis and number theory. Some of these problems are the following: The Integer Chebyshev Problem. Find a nonzero polynomial of degree n with integer eoeffieients that has smallest possible supremum norm on the unit interval. Littlewood's Problem. Find a polynomial of degree n with eoeffieients in the set { + 1, -I} that has smallest possible supremum norm on the unit disko The Prouhet-Tarry-Escott Problem. Find a polynomial with integer co­ effieients that is divisible by (z - l)n and has smallest possible 1 norm. (That 1 is, the sum of the absolute values of the eoeffieients is minimal.) Lehmer's Problem. Show that any monie polynomial p, p(O) i- 0, with in­ teger coefficients that is irreducible and that is not a cyclotomic polynomial has Mahler measure at least 1.1762 .... All of the above problems are at least forty years old; all are presumably very hard, certainly none are completely solved; and alllend themselves to extensive computational explorations. The techniques for tackling these problems are various and include proba­ bilistic methods, combinatorial methods, "the circle method," and Diophantine and analytic techniques. Computationally, the main tool is the LLL algorithm for finding small vectors in a lattice. The book is intended as an introduction to a diverse collection of techniques.

Reviews

From the reviews of the first edition:

MAA ONLINE

"Explanations are thorough but not easy to understand. Nevertheless, they can be understood by the determined graduate student in mathematics. However, the ideal mix would be a collection of mathematics and computer science students, as the level of computer expertise needed to code the solutions to the problems is at the upper division level…Research mathematicians often need to be able to write code to attack specific problems when no appropriate software tool is available. This book is ideal for a course designed to teach graduate students how to do that as long as they have or can obtain the necessary programming knowledge."

P. Borwein

Computational Excursions in Analysis and Number Theory

"Borwein has collected known results in the direction of several related, appealing, old, open problems (Integer Chebyshev, Prouhet-Tarry-Escott, Erdos-Szekeres, Littlewood). Far from narrow, this interdisciplinary book draws on Diophantine, analytic, and probabilistic techniques. Also, by dint of the celebrated lattice reduction algorithm, some aspects of these problems admit attack by computer; a handful of intriguing computer graphics offer visceral evidence of the intrinsic complexity of the underlying phenomena. Pisot and Salam numbers make terrific enrichment material for undergraduates. As in all Borwein's books, we get beautiful mathematics gracefully explained."—CHOICE

"This extraordinary book brings together a variety of old problems – old, but very much alive - about polynomials with integer co-efficients. … The necessary background is also presented, which makes the book self-contained … . this book is suitable for advanced students of analysis and analytic number theory. It is very well written, rather concise and to the point. … Strongly recommended for specialists in computational analysis and number theory." (R. Stroeker, NieuwArchief voor Wiskunde, Vol. 7 (3), 2006)

Authors and Affiliations

  • Department of Mathematics and Statistics, Simon Fraser University, Burnaby, Canada

    Peter Borwein

Bibliographic Information

  • Book Title: Computational Excursions in Analysis and Number Theory

  • Authors: Peter Borwein

  • Series Title: CMS Books in Mathematics

  • DOI: https://doi.org/10.1007/978-0-387-21652-2

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 2002

  • Hardcover ISBN: 978-0-387-95444-8Published: 12 July 2002

  • Softcover ISBN: 978-1-4419-3000-2Published: 03 December 2010

  • eBook ISBN: 978-0-387-21652-2Published: 06 December 2012

  • Series ISSN: 1613-5237

  • Series E-ISSN: 2197-4152

  • Edition Number: 1

  • Number of Pages: X, 220

  • Number of Illustrations: 4 b/w illustrations

  • Topics: Number Theory, Analysis, Symbolic and Algebraic Manipulation

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