Monic integer Chebyshev problem
Authors:
P. B. Borwein, C. G. Pinner and I. E. Pritsker
Journal:
Math. Comp. 72 (2003), 19011916
MSC (2000):
Primary 11C08; Secondary 30C10
Published electronically:
January 8, 2003
MathSciNet review:
1986811
Fulltext PDF Free Access
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Abstract: We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let denote the monic polynomials of degree with integer coefficients. A monic integer Chebyshev polynomial satisfies
and the monic integer Chebyshev constant is then defined by This is the obvious analogue of the more usual integer Chebyshev constant that has been much studied. We compute for various sets, including all finite sets of rationals, and make the following conjecture, which we prove in many cases. Conjecture. Suppose is an interval whose endpoints are consecutive Farey fractions. This is characterized by Then
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
Email:
pinner@math.ksu.edu
I. E. Pritsker
Affiliation:
Department of Mathematics, 401 Mathematical Sciences, Oklahoma State University, Stillwater, Oklahoma 74078
Email:
igor@math.okstate.edu
DOI:
http://dx.doi.org/10.1090/S0025571803014777
PII:
S 00255718(03)014777
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 EPS9874732 and matching support from the state of Kansas (Pinner), and NSF grant DMS9996410 (Pritsker).
Article copyright:
© Copyright 2003
by the authors
