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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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The integer Chebyshev problem
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by Peter Borwein and Tamás Erdélyi PDF
Math. Comp. 65 (1996), 661-681 Request permission


We are concerned with the problem of minimizing the supremum norm on an interval of a nonzero polynomial of degree at most $n$ with integer coefficients. This is an old and hard problem that cannot be exactly solved in any nontrivial 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 intevals $[0, \delta ]$ and for small degrees $d$, $x^{d}$ achieves the minimal norm. There is a natural conjecture, due to the Chudnovskys and others, as to what the “integer transfinite diameter” of $[0,1]$ should be. We show that this conjecture is false. The problem is then related to a trace problem for totally positive algebraic integers due to Schur and Siegel. Several open problems are raised.
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Additional Information
  • Peter Borwein
  • Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6
  • Email:
  • Tamás Erdélyi
  • Affiliation: Department of Mathematics, Texas A & M University, College Station, Texas 77843
  • Email:
  • Received by editor(s): April 25, 1994
  • Received by editor(s) in revised form: September 5, 1994, and February 12, 1995
  • Additional Notes: Research supported in part by NSERC of Canada.
  • © Copyright 1996 American Mathematical Society
  • Journal: Math. Comp. 65 (1996), 661-681
  • MSC (1991): Primary 11J54, 11B83
  • DOI:
  • MathSciNet review: 1333305