# Computational Excursions in Analysis and Number Theory

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

### Popular passages

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