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Monic integer Chebyshev problem

Authors: P. B. Borwein, C. G. Pinner and I. E. Pritsker
Journal: Math. Comp. 72 (2003), 1901-1916
MSC (2000): Primary 11C08; Secondary 30C10
Published electronically: January 8, 2003
MathSciNet review: 1986811
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Abstract: We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let ${\mathcal{M}}_n({\mathbb{Z} })$ denote the monic polynomials of degree $n$ with integer coefficients. A monic integer Chebyshev polynomial $M_n \in {\mathcal{M}}_n({\mathbb{Z} })$satisfies

\begin{displaymath}\Vert M_n \Vert _{E} = \inf_{P_n \in{\mathcal{M}}_n ( {\mathbb{Z} })} \Vert P_n \Vert _{E}. \end{displaymath}

and the monic integer Chebyshev constant is then defined by

\begin{displaymath}t_M(E) := \lim_{n \rightarrow \infty} \Vert M_n \Vert _{E}^{1/n}. \end{displaymath}

This is the obvious analogue of the more usual integer Chebyshev constant that has been much studied.

We compute $t_M(E)$ for various sets, including all finite sets of rationals, and make the following conjecture, which we prove in many cases.

Conjecture. Suppose $[{a_2}/{b_2},{a_1}/{b_1}]$ is an interval whose endpoints are consecutive Farey fractions. This is characterized by $a_1b_2-a_2b_1=1.$ Then

\begin{displaymath}t_M[{a_2}/{b_2},{a_1}/{b_1}] = \max(1/b_1,1/b_2).\end{displaymath}

This should be contrasted with the nonmonic integer Chebyshev constant case, where the only intervals for which the constant is exactly computed are intervals of length 4 or greater.

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Additional Information

P. B. Borwein
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada

C. G. Pinner
Affiliation: Department of Mathematics, 138 Cardwell Hall, Kansas State University, Manhattan, Kansas 66506

I. E. Pritsker
Affiliation: Department of Mathematics, 401 Mathematical Sciences, Oklahoma State University, Stillwater, Oklahoma 74078

Keywords: Chebyshev polynomials, integer Chebyshev constant, integer transfinite diameter.
Received by editor(s): August 29, 2001
Received by editor(s) in revised form: December 20, 2001
Published electronically: January 8, 2003
Additional Notes: Research of the authors was supported in part by the following grants: NSERC of Canada and MITACS (Borwein), NSF grant EPS-9874732 and matching support from the state of Kansas (Pinner), and NSF grant DMS-9996410 (Pritsker).
Article copyright: © Copyright 2003 by the authors