Advertisement

Mathematical Notes

, Volume 105, Issue 1–2, pp 291–300 | Cite as

Chebyshev Polynomials and Integer Coefficients

  • R. M. TrigubEmail author
Article
  • 9 Downloads

Abstract

Generalized Chebyshev polynomials are introduced and studied in this paper. They are applied to obtain a lower bound for the sup-norm on the closed interval for nonzero polynomials with integer coefficients of arbitrary degree.

Keywords

extremal properties of polynomials Hilbert–Fekete theorem integer algebraic numbers asymptotic law of the distribution of primes Eisenstein criterion for the irreducibility of polynomials 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. M. Goluzin, Geometric Theory of Functions of a Complex Variable (Gostekhizdat, Moscow–Leningrad, 1952) [in Russian].zbMATHGoogle Scholar
  2. 2.
    T. J. Rivlin, Chebyshev Polynomials. FromApproximation Theory to Algebra and Number Theory (John Wiley & Sons, New York, 1990).zbMATHGoogle Scholar
  3. 3.
    P. B. Borwein, C. G. Pinner, and I. E. Pinsker, “Monic integer Chebyshev problem,” Math. Comp. 72 (244), 1901–1916 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    M. Fekete, “Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten,” Math. Z. 17, 228–249 (1923).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    H. Hewilt and H. S. Zuckerman, “Approximation by polynomials with integral coefficients, a reformulation of the Stthe one–Weierstrass theorem,” Duke Math. J. 26, 305–324 (1959).MathSciNetCrossRefGoogle Scholar
  6. 6.
    M. Fekete, “Approximation par polynomes over conditions diophantions,” C. R. Acad. Sci. Paris 239, 1337–1339 (1954).MathSciNetzbMATHGoogle Scholar
  7. 7.
    Y. Okada, “On approximate polynomials with integral coefficients only,” TôhokuMath. J. 23, 26–35 (1923).zbMATHGoogle Scholar
  8. 8.
    J. Cassels, An Introduction to Diophantine Approximation (Cambridge Univ. Press, Cambridge, 1957; Inostr. Lit.,Moscow, 1961).zbMATHGoogle Scholar
  9. 9.
    R. M. Trigub, “Approximation of functions under Diophantine conditions by polynomials with integer coefficients,” in Metric Questions of the Theory of Functions and Mappings (Naukova Dumka, Kiev, 1971), Issue 2, pp. 267–353 [in Russian].Google Scholar
  10. 10.
    F. Amoroso, “Sur le diamétre transfini entier d’un intervalle réel,” Ann. Inst. Fourier (Grenoble) 40 (4), 885–911 (1991).CrossRefzbMATHGoogle Scholar
  11. 11.
    B. S. Kashin, “Algebraic polynomials with integer coefficients deviating little from zero on an interval,” Mat. Zametki 50 (3), 58–67 (1991) [Math. Notes 50 (3), 921–927 (1991)].MathSciNetzbMATHGoogle Scholar
  12. 12.
    R. M. Trigub, “Approximation of smooth functions and constants by polynomials with integer and natural coefficients,” Mat. Zametki 70 (1), 123–136 (2001) [Math. Notes 70 (1), 110–122 (2001)].MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    A. O. Gelfond, “Comments on the papers,” in P. L. Chebyshev, Collected Works, Vol. 1: Number Theory (Moscow–Leningrad, 1944), pp. 285–288 [in Russian].Google Scholar
  14. 14.
    D. S. Gorshkov, “On deviations from zero of polynomials with integer rational coefficients on the interval [0, 1],” in Proceedings of the III All–Union Mathematical Congress (AN SSSR, Moscow, 1959), Vol. 4, pp. 5–7 [in Russian].Google Scholar
  15. 15.
    H. I. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis (Amer. Math. Soc., Providence, RI, 1994).CrossRefzbMATHGoogle Scholar
  16. 16.
    K. G. Hare, “Generalized Gorshkov–Wirsing polynomials and the integer Chebyshev problem,” Exp. Math. 20 (2), 189–200 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    P. Borwein and T. Erdélyi, “The integer Chebyshev problem,” Math. Comp. 65 (214), 661–681 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    V. V. Prasolov, Polynomials (MTsNMO,Moscow, 2003) [in Russian].Google Scholar
  19. 19.
    I. E. Pritsker, “Small polynomials with integer coefficients,” J. Anal. Math. 96, 151–190 (2005).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Sumy State UniversitySumyUkraine

Personalised recommendations