## 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. |

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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 |

### Common terms and phrases

algebraic integer algebraic number asymptotic Barker polynomials bn-i Chebyshev polynomials column vector Computational Problems Cl conjecture Corollary cyclotomic polynomials defined equivalent Erdos Fekete polynomials full reductions Golay complementary pair Golay pairs height 1 polynomials HJLS ideal solution inequality integer Chebyshev problem integer coefficients integer relation Introductory Exercises irreducible L4 norm lattice Legendre symbol Lemma Littlewood polynomials LLL algorithm lm(q lower bound Mahler measure Math merit factor minimal monic polynomial multiplicity nonzero nth root odd coefficients odd prime Pisot number pn(z polynomial of degree polynomials pn polynomials with integer positive constant positive integers Problems from Chapter product of cyclotomic Proof Prouhet-Tarry-Escott problem Prove PSLQ algorithm real zeros reciprocal polynomials Research Problems Rl result roots of unity Rudin-Shapiro polynomials Salem number satisfies Selected References sequence Show skewsymmetric smallest possible Suppose supremum norm Theorem unit circle unit disk upper bound values vanishing Waring problem zero of order Zp[z

### 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.