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The Integer Chebyshev Problem

Borwein, Peter and Erdelyi, Tamas (1994) The Integer Chebyshev Problem. [Preprint]

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      Abstract

      We are concerned with the problem of minimizing the supremum norm on an interval of a non-zero polynomial of degree at most $n$ with integer coefficients. This is an old and hard problem that cannot be exactly solved in any non-trivial cases. We examine the case of the interval [0,1] in most detail. Here we improve the known bounds a small but interesting amount. This allows us to garner further information about the structure of such minimal polynomials and their factors. This is primarily a (substantial) computational exercise. We also examine some of the structure of such minimal ``integer Chebyshev'' polynomials, showing for example, that on small intervals $[0, \delta]$ and for small degrees $d$, $x^d$ achieves the minimal norm. The problem is then related to a trace problem for totally positive algebraic integers due to Schur and Siegel. Several open problems are raised.

      Item Type: Preprint
      Additional Information: pubdom FALSE
      Uncontrolled Keywords: transfinite diameter, integers, diophantine approximation, trace, Chebyshev, polynomial
      Subjects: 11-xx Number theory > 11Bxx Sequences and sets
      11-xx Number theory > 11Jxx Diophantine approximation, transcendental number theory
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 17 Nov 2003
      Last Modified: 21 Apr 2010 11:13
      URI: https://docserver.carma.newcastle.edu.au/id/eprint/83

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