Computational Excursions in Analysis and Number Theory

Peter Borwein

ISBN 10: 0387954449 / ISBN 13: 9780387954448

Verlag: Springer New York, Springer US, 2002

Beschreibung

Druck auf Anfrage Neuware - Printed after ordering - 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.

Bestandsnummer des Verkäufers 9780387954448

Bibliografische Details

• Titel: Computational Excursions in Analysis and Number Theory
• Verlag: Springer New York, Springer US
• Erscheinungsjahr: 2002
• Sprache: Englisch
• ISBN-10: 0387954449
• ISBN-13: 9780387954448
• Einband: Buch
• Zustand: Neu
• Abmessungen: 241B x 160 H x 18 T Millimeter Breite: 241 mm, Höhe: 160 mm, Tiefe: 18 mm
• Gewicht: 524 Gramm

Über diesen Titel

Inhaltsangabe

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.

Von der hinteren Coverseite

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 Americas Chauvenet Prize and the Merten M. Hasse Prize for expository writing in mathematics.