Mathematical Notes

, Volume 64, Issue 4, pp 440–449 | Cite as

On the linear independence of numbers over number fields

  • E. V. Bedulev


In the present paper, the problem of a lower bound for the measure of linear independence of a given collection of numbersθ1, …,θ n is considered under the assumption that, for a sequence of polynomials whose coefficients are algebraic integers, upper and lower estimates at the point (θ1, …,θ n ) are known. We use a method that generalizes the Nesterenko method to the case of an arbitrary algebraic number field.

Key words

number field degree of transcendence measure of transcendence 


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  1. 1.
    G. V. Chudnovskii [G. V. Choodnovsky],Certain analytic methods in the theory of transcendental numbers [in Russian], Preprint IM-74-8,Analytic methods in Diophantine approximations [in Russian], Preprint IM-74-9, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev (1974).Google Scholar
  2. 2.
    E. Ressat, “Un critère d'indépendance algébrique,”J. Reine Angew. Math.,329, 66–81 (1981).MathSciNetGoogle Scholar
  3. 3.
    M. Waldschmidt and Zhu Yao Chen, “Une généralisation en plusieures variables d'un critère de transcendance de Gelfond,”C. R. Acad. Sci. Paris. Sér. I. Math.,297, 229–232 (1983).Google Scholar
  4. 4.
    Yu. V. Nesterenko, “A sufficient condition for algebraic independence of numbers,”Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.], No. 4, 63–68 (1983).MATHMathSciNetGoogle Scholar
  5. 5.
    P. Phillipon, “Critères pour l'indépendance algébrique de familles de nombres,”C R. Math. Rep. Acad. Sci. Canada,6, 285–290 (1984).MathSciNetGoogle Scholar
  6. 6.
    Yu. V. Nesterenko, “Linear independence of numbers,”Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.], No. 1, 46–49 (1985).MATHMathSciNetGoogle Scholar
  7. 7.
    R. Gantmakher,The Theory of Matrices [in Russian], Nauka, Moscow (1986).Google Scholar
  8. 8.
    S. Lang,Fundamentals of Diophantine Geometry, Springer-Verlag, New York-Berlin (1986).Google Scholar
  9. 9.
    V. F. Kagan,Foundations of the Theory of Determinants [in Russian], Gos. Izd. Ukrainy, Odessa (1921).Google Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 1999

Authors and Affiliations

  • E. V. Bedulev
    • 1
  1. 1.M. V. Lomonosov Moscow State UniversityUSSR

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