Computational Excursions in Analysis and Number TheoryThis 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. |
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 |
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Common terms and phrases
algebraic integer algebraic number Amer asymptotic Barker polynomials Chebyshev polynomials column vector Comp conjecture cyclotomic polynomials defined denote Erdős Fekete polynomials full reductions Golay complementary pair Golay pairs height 1 polynomial HJLS ideal solution II(k inequality integer coefficients integer relation Introductory Exercises E1 irreducible iteration L4 norm lattice Legendre symbol Lemma Littlewood polynomials LLL algorithm London Math lower bound Mahler measure merit factor minimal modulus monic polynomial multiplicity nonzero nth root Number Theory odd coefficients odd prime Pisot number Pn(z polynomial in Ln polynomial of degree polynomials with integer positive constant Problems from Chapter product of cyclotomic Proof Prouhet-Tarry-Escott problem PSLQ algorithm reciprocal polynomials reduced basis result roots of unity Rudin-Shapiro polynomials Salem numbers satisfies Show Suppose supremum norm Theorem unit circle unit disk values vanishing Waring problem
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.