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

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: 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.

References [Enhancements On Off] (What's this?)

• Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR 1367960
• Peter Borwein and Tamás Erdélyi, The integer Chebyshev problem, Math. Comp. 65 (1996), no. 214, 661–681. MR 1333305, DOI https://doi.org/10.1090/S0025-5718-96-00702-8
• G. V. Chudnovsky, Number theoretic applications of polynomials with rational coefficients defined by extremality conditions, Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, pp. 61–105. MR 717590
• M. Fekete, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Zeit. 17 (1923), 228-249.
• M. Fekete, Über den tranfiniten Durchmesser ebener Punktmengen II, Math. Zeit. 32 (1930), 215-221.
• Le Baron O. Ferguson, Approximation by polynomials with integral coefficients, Mathematical Surveys, vol. 17, American Mathematical Society, Providence, R.I., 1980. MR 560902
• V. Flammang, G. Rhin, and C. J. Smyth, The integer transfinite diameter of intervals and totally real algebraic integers, J. Théor. Nombres Bordeaux 9 (1997), no. 1, 137–168 (English, with English and French summaries). MR 1469665
• G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR 0247039
• Laurent Habsieger and Bruno Salvy, On integer Chebyshev polynomials, Math. Comp. 66 (1997), no. 218, 763–770. MR 1401941, DOI https://doi.org/10.1090/S0025-5718-97-00829-6
• D. Hilbert, Ein Beitrag zur Theorie des Legendreschen Polynoms, Acta Math. 18 (1894), 155-159.
• Hugh L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, vol. 84, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR 1297543
• Y. Okada, On approximate polynomials with integral coefficients only, Tohoku Math. J. 23 (1924), 26-35.
• I. E. Pritsker, Small polynomials with integer coefficients, in press.
• Thomas Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR 1334766
• Theodore J. Rivlin, Chebyshev polynomials, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1990. From approximation theory to algebra and number theory. MR 1060735
• R. M. Trigub, Approximation of functions with Diophantine conditions by polynomials with integral coefficients, Metric questions of the theory of functions and mappings, No. 2 (Russian), Izdat. “Naukova Dumka”, Kiev, 1971, pp. 267–333 (Russian). MR 0312121
• M. Tsuji, Potential theory in modern function theory, Chelsea Publishing Co., New York, 1975. Reprinting of the 1959 original. MR 0414898

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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).