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Izv. Akad. Nauk SSSR Ser. Mat., 1971, Volume 35, Issue 5, Pages 973–990 (Mi izv2114)  

This article is cited in 5 scientific papers (total in 6 papers)

An effective refinement of the exponent in Liouville's theorem

N. I. Fel'dman


Abstract: For every algebraic number $\alpha$ of degree $n\geq3$ there exist effective positive constants $a$ and $C$ such that for any rational integers $q>0$ and $p$ we have
$$ |\alpha-\frac pq|>Cq^{a-n}. $$
We also derive an effective boundary of the type $C_1m^{a_1}$ for the solutions of the Diophantine equation $f(x,y)=m$, where $f$ is a form of degree $\geq3$.

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English version:
Mathematics of the USSR-Izvestiya, 1971, 5:5, 985–1002

Bibliographic databases:

UDC: 511.6
MSC: 12B15, 10B99, 10F25
Received: 18.02.1971

Citation: N. I. Fel'dman, “An effective refinement of the exponent in Liouville's theorem”, Izv. Akad. Nauk SSSR Ser. Mat., 35:5 (1971), 973–990; Math. USSR-Izv., 5:5 (1971), 985–1002

Citation in format AMSBIB
\Bibitem{Fel71}
\by N.~I.~Fel'dman
\paper An effective refinement of the exponent in Liouville's theorem
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1971
\vol 35
\issue 5
\pages 973--990
\mathnet{http://mi.mathnet.ru/izv2114}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=289418}
\zmath{https://zbmath.org/?q=an:0237.10018}
\transl
\jour Math. USSR-Izv.
\yr 1971
\vol 5
\issue 5
\pages 985--1002
\crossref{https://doi.org/10.1070/IM1971v005n05ABEH001130}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. V. Kotov, V. G. Sprindzhuk, “The Thue–Mahler equation in a relative field and approximation of algebraic numbers by algebraic numbers”, Izv. Math., 41:4 (1977), 677–707  mathnet  crossref  mathscinet  zmath
    2. V. G. Sprindzhuk, “Achievements and problems in Diophantine approximation theory”, Russian Math. Surveys, 35:4 (1980), 1–80  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. H. Luckhardt, “Herbrand-Analysen zweier Beweise des Satzes von Roth: Polynomiale Anzahlschranken”, J. symb. log, 54:01 (1989), 234  crossref
    4. N. M. Korobov, Yu. V. Nesterenko, A. B. Shidlovskii, “Naum Il'ich Feld'man (obituary)”, Russian Math. Surveys, 50:6 (1995), 1247–1252  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. E. M. Matveev, “An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers”, Izv. Math., 62:4 (1998), 723–772  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. Bugeaud Y., “Linear Forms in Logarithms and Applications”, Linear Forms in Logarithms and Applications, Irma Lectures in Mathematics and Theoretical Physics, 28, Eur. Math. Soc., 2018, 1–224  crossref  isi
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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