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

DOI:
https://doi.org/10.1090/S0025-5718-03-01477-7

Published electronically:
January 8, 2003

MathSciNet review:
1986811

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Abstract | References | Similar Articles | Additional Information

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{equation*} \| M_n \|_{E} = \inf _{P_n \in \mathcal {M}_n ( {\mathbb {Z}})} \| P_n \|_{E}. \end{equation*} and the *monic integer Chebyshev constant* is then defined by \begin{equation*} t_M(E) := \lim _{n \rightarrow \infty } \| M_n \|_{E}^{1/n}. \end{equation*} 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{equation*}t_M[{a_2}/{b_2},{a_1}/{b_1}] = \max (1/b_1,1/b_2).\end{equation*} 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

Email:
pborwein@cecm.sfu.ca

**C. G. Pinner**

Affiliation:
Department of Mathematics, 138 Cardwell Hall, Kansas State University, Manhattan, Kansas 66506

MR Author ID:
319822

Email:
pinner@math.ksu.edu

**I. E. Pritsker**

Affiliation:
Department of Mathematics, 401 Mathematical Sciences, Oklahoma State University, Stillwater, Oklahoma 74078

MR Author ID:
319712

Email:
igor@math.okstate.edu

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