This book is designed for a computationally intensive graduate course based around a collection of classical unsolved extremal problems for polynomials. These problems, all of which lend themselves to extensive
computational exploration, live at the interface of analysis,
combinatorics and number theory so the techniques involved are diverse.A main computational tool used is the LLL algorithm for finding small vectors in a lattice.
Many exercises and open research problems are included. Indeed one aim of the book is to tempt the able reader into the rich possibilities for research in this area.
Peter Borwein is Professor of Mathematics at Simon Fraser University and the Associate Director of the Centre for Experimental and Constructive Mathematics. He is also the recipient of the Mathematical Association of America's Chauvenet Prize and the Merten M. Hasse Prize for expository
writing in mathematics. This introduction to computational number theory is centered on a number of problems that live at the interface of analytic, computational and Diophantine number theory, and provides a diverse collection of techniques for solving number- theoretic problems. There are many exercises and open research problems included.
Preface Introduction LLL and PSLQ Pisot and Salem Numbers Rudin-Shapiro Polynomials Fekete Polynomials Products of Cyclotomic Polynomials Location of Zeros Maximal Vanishing Diophantine Approximation of Zeros The Integer-Chebyshev Problem The Prouhet-Tarry-Escott Problem The Easier Waring Problem The Erdös-Szekeres Problem Barker Polynomials and Golay Pairs The Littlewood Problem Spectra Appendix A: A Compendium of Inequalities B: Lattice Basis Reduction and Integer Relations C: Explicit Merit Factor Formulae D: Research Problems References Index
From the reviews of the first edition:
"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."
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, Nieuw Archief voor Wiskunde, Vol. 7 (3), 2006)
* Preface * Introduction * LLL and PSLQ * Pisot and Salem Numbers * Rudin-Shapiro Polynomials * Fekete Polynomials * Products of Cyclotomic Polynomials * Location of Zeros * Maximal Vanishing * Diophantine Approximation of Zeros * The Integer-Chebyshev Problem * The Prouhet-Tarry-Escott Problem * The Easier Waring Problem * The Erdös-Szekeres Problem * Barker Polynomials and Golay Pairs * The Littlewood Problem * Spectra * Appendix A: A Compendium of Inequalities * B: Lattice Basis Reduction and Integer Relations * C: Explicit Merit Factor Formulae * D: Research Problems * References * Index
This introduction to computational number theory is centered on a number of problems that live at the interface of analytic, computational and Diophantine number theory, and provides a diverse collection of techniques for solving number- theoretic problems. There are many exercises and open research problems included.
This book is designed for a course in computational number theory. It is based around a number of difficult old problems that live at the interface of analytic, computational and Diophantine number theory. The techniques for tackling these problems are various and include probabilistic methods, combinatorial methods, Diophantine and analytic techniques. The main computational tool used is the LLL algorithm for finding small vectors in a lattice. The book is intended as an introduction to a diverse collection of techniques for solving number- theoretic problems. For all chapters, the author has suggested related research papers where additional details may be pursued. There are many exercises and open research problems included. Indeed