Computational Excursions in Analysis and Number Theory

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Springer Science & Business Media, Jul 12, 2002 - Computers - 220 pages
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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.
  

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Contents

Introduction
1
LLL and PSLQ
11
Pisot and Salem Numbers
15
RudinShapiro Polynomials
27
Fekete Polynomials
37
Products of Cyclotomic Polynomials
43
Location of Zeros
53
Maximal Vanishing
59
The Easier Waring Problem
97
The ErdosSzekeres Problem
103
Barker Polynomials and Golay Pairs
109
The Littlewood Problem
121
Spectra
133
A Compendium of Inequalities
141
Lattice Basis Reduction and Integer Relations
153
Explicit Merit Factor Formulae
181

Diophantine Approximation of Zeros
67
The Integer Chebyshev Problem
75
The ProuhetTarryEscott Problem
85
Research Problems
195
Index
217
Copyright

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Page 212 - Proof of a conjecture of P. Erdos on the derivative of a polynomial, Bull. Amer. Math. Soc. 50 (1944), 509-513.

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