Some old problems on polynomials with integer coefficients.

*(English)*Zbl 0930.12001
Chui, Charles K. (ed.) et al., Approximation theory IX. Volume 1. Theoretical aspects. Proceedings of the 9th international conference, Nashville, TN, USA, January 3–6, 1998. Nashville, TN: Vanderbilt University Press. Innovations in Applied Mathematics. 31-50 (1998).

Author’s abstract: We survey a number of old and difficult problems all of which involve finding polynomials with integer coefficients and small norm. These problems include: the Integer Chebyshev problem of Hilbert and Fekete; the Prouhet-Tarry-Escott problem; various conjectures of Littlewood and various conjectures of Erdős. These problems are unsolved and most are at least 35 years old. They do however lend themselves to partial solution and one suspects that they are not, in fact, totally intractable. They are also all amenable to being computed on and offer some interesting computational challenges.

For the entire collection see [Zbl 0910.00046].

For the entire collection see [Zbl 0910.00046].

Reviewer: Edward J.Barbeau (Toronto)

##### MSC:

12-02 | Research exposition (monographs, survey articles) pertaining to field theory |

12D99 | Real and complex fields |

11C08 | Polynomials in number theory |

30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |

26C10 | Real polynomials: location of zeros |

12D10 | Polynomials in real and complex fields: location of zeros (algebraic theorems) |

41A10 | Approximation by polynomials |

11D41 | Higher degree equations; Fermat’s equation |