Computational Excursions in Analysis and Number Theory (eBook)

Peter Borwein (Autor)

eBook Download: PDF

2012
Springer New York (Verlag)
978-0-387-21652-2 (ISBN)

Infos

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, "e;the circle method,"e; 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.

Produkt-Details

• Erscheint lt. Verlag: 6.12.2012
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• ISBN-10: 0-387-21652-9 / 0387216529
• ISBN-13: 978-0-387-21652-2 / 9780387216522