Computational Excursions in Analysis and Number Theory
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Computational Excursions in Analysis and Number Theory

Computational Excursions in Analysis and Number Theory


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About the Book

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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Product Details
  • ISBN-13: 9780387954448
  • Publisher: Springer
  • Publisher Imprint: Springer
  • Depth: 13
  • Height: 243 mm
  • No of Pages: 220
  • Series Title: CMS Books in Mathematics
  • Weight: 557 gr
  • ISBN-10: 0387954449
  • Publisher Date: 12 Jul 2002
  • Binding: Hardback
  • Edition: 2002 ed.
  • Language: English
  • Returnable: N
  • Spine Width: 15 mm
  • Width: 161 mm


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