Order function for almost all numbers

  • Yu. V. Nesterenko


For almost all pointsξ∃ Rm (m≥2) the inequality
$$\sup {\text{ }}ln \frac{1}{{|P (\xi )|}} \ll (ln u)^{m + 2} ,$$
is valid, where the upper bound is taken over all nonzero polynomials P for which
$$\exp {\text{ }}(\deg {\text{ }}P){\text{ }}L{\text{ }}(P) \leqslant u,$$
where L(P) is the sum of the moduli of the coefficients of P. When m=1 the exponent of the right side is equal to 2.


Order Function Nonzero Polynomial 
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Literature cited

  1. 1.
    K. Mahler, “On the order function of transcendental numbers,” Acta arithm.,18, 63–76 (1971).Google Scholar
  2. 2.
    A. O. Gel'fond, Transcendental and Algebraic Numbers, Dover, New York (1960).Google Scholar
  3. 3.
    N. I. Fel'dman, “Approximation of some transcendental numbers, I,” Izv. Akad. Nauk SSSR, Ser. Matem.,15, No. 1, 53–74 (1951).Google Scholar
  4. 4.
    S. Lang, Introduction to Transcendental Numbers, Addison-Wesley, Reading, Mass. (1966).Google Scholar
  5. 5.
    Yu. A. Brudnyi and M. I. Ganzburg, “On an extremal problem for polynomials in n variables,” Izv. Akad. Nauk SSSR, Ser. Matem.,37, No. 2, 344–355 (1973).Google Scholar

Copyright information

© Consultants Bureau 1974

Authors and Affiliations

  • Yu. V. Nesterenko
    • 1
  1. 1.M. V. Lomonosov Moscow State UniversityUSSR

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