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Mat. Sb. (N.S.), 1974, Volume 95(137), Number 3(11), Pages 396–417 (Mi msb3760)  

This article is cited in 15 scientific papers (total in 15 papers)

Estimates from below of polynomials in the values of analytic functions of a certain class

A. I. Galochkin

Abstract: Estimates from below are obtained for polynomials with integral coefficients in the values of certain Siegel $G$-functions at the algebraic points of a special form. In particular, it is proved that if $\alpha_1,…,\alpha_s$ ($\alpha_1\cdots\alpha_s\ne0$) are pairwise distinct algebraic numbers, $q$ is a natural number, and $P(x_1,…,x_s)\not\equiv0$ is a polynomial with integral coefficients of degree not greater than $d$ and height not exceeding $H$, then for $q>q_0(d,\alpha_1,…,\alpha_s)$ we have
$$|P(\ln(1+\frac{\alpha_1}q),…,\ln(1+\frac{\alpha_s}q))|>q^{-\lambda}H^{-\mu}, $$
where the constants $q_0$ and $\mu$ can be effectively computed.
Bibliography: 17 titles.

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English version:
Mathematics of the USSR-Sbornik, 1974, 24:3, 385–407

Bibliographic databases:

UDC: 511.8
MSC: 33A35, 12A20, 10F25
Received: 17.05.1973

Citation: A. I. Galochkin, “Estimates from below of polynomials in the values of analytic functions of a certain class”, Mat. Sb. (N.S.), 95(137):3(11) (1974), 396–417; Math. USSR-Sb., 24:3 (1974), 385–407

Citation in format AMSBIB
\by A.~I.~Galochkin
\paper Estimates from below of polynomials in the values of analytic functions of a~certain class
\jour Mat. Sb. (N.S.)
\yr 1974
\vol 95(137)
\issue 3(11)
\pages 396--417
\jour Math. USSR-Sb.
\yr 1974
\vol 24
\issue 3
\pages 385--407

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    This publication is cited in the following articles:
    1. V. G. Chirskii, “Ob arifmeticheskikh svoistvakh znachenii ellipticheskikh integralov”, UMN, 32:1(193) (1977), 211–212  mathnet  mathscinet  zmath
    2. E. M. Nikishin, “On irrationality of the values of the functions $F(x,s)$”, Math. USSR-Sb., 37:3 (1980), 381–388  mathnet  crossref  mathscinet  zmath  isi
    3. E. M. Matveev, “Linear forms in the values of $G$-functions, and Diophantine equations”, Math. USSR-Sb., 45:3 (1983), 379–396  mathnet  crossref  mathscinet  zmath
    4. L. A. Gutnik, “The linear independence over $\mathbb Q$ of dilogarithms at rational points”, Russian Math. Surveys, 37:5 (1982), 176–177  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. Chudnovsky D., Chudnovsky G., “Pade Approximations to Solutions of Linear-Differential Equations and Applications to Diophantine Analysis”, 1052, 1984, 85–167  mathscinet  zmath  isi
    6. Xu Guangshan, “On the arithmetic properties of the values ofG-functions”, Acta Math Sinica, 1:2 (1985), 141  crossref  mathscinet  zmath
    7. V. N. Sorokin, “On the irrationality of the values of hypergeometric functions”, Math. USSR-Sb., 55:1 (1986), 243–257  mathnet  crossref  mathscinet  zmath
    8. Vaananen K., Xu G., “On Linear-Forms of G-Functions”, Acta Arith., 50:3 (1988), 251–263  crossref  mathscinet  zmath  isi
    9. Vaananen K., Xu G., “On the Arithmetic Properties of the Values of Gamma-Functions”, J. Aust. Math. Soc. A-Pure Math. Stat., 47:Part 1 (1989), 71–82  crossref  mathscinet  zmath  isi
    10. W. V. Zudilin, “On a measure of irrationality for values of $G$-functions”, Izv. Math., 60:1 (1996), 91–118  mathnet  crossref  crossref  mathscinet  zmath  isi
    11. W. V. Zudilin, “Cancellation of factorials”, Sb. Math., 192:8 (2001), 1181–1207  mathnet  crossref  crossref  mathscinet  zmath  isi
    12. Lagarias J., “On the Normality of Arithmetical Constants”, Exp. Math., 10:3 (2001), 355–368  crossref  mathscinet  zmath  isi
    13. V. N. Sorokin, “Estimates for polynomials in logarithms of some rational numbers”, J. Math. Sci., 146:2 (2007), 5759–5770  mathnet  crossref  mathscinet  zmath  elib
    14. V. G. Lysov, “Ob approksimatsiyakh Ermita–Pade dlya proizvedeniya dvukh logarifmov”, Preprinty IPM im. M. V. Keldysha, 2017, 141, 24 pp.  mathnet  crossref
    15. V. G. Lysov, “O diofantovykh priblizheniyakh proizvedeniya logarifmov”, Preprinty IPM im. M. V. Keldysha, 2018, 158, 20 pp.  mathnet  crossref  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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