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Mat. Sb. (N.S.), 1984, Volume 123(165), Number 4, Pages 435–459 (Mi msb2030)  

This article is cited in 20 scientific papers (total in 20 papers)

On algebraic independence of algebraic powers of algebraic numbers

Yu. V. Nesterenko

Abstract: It is proved that among the numbers $\alpha^\beta,\alpha^{\beta^2},…,\alpha^{\beta^{d-1}}$, where $\alpha$ is algebraic, $\alpha\ne0,1$ and $\beta$ is algebraic of degree $d\geqslant2$, there are no fewer than $[\log_2(d+1)]$ which are algebraically independent over $\mathbf Q$.
Bibliography: 17 titles.

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English version:
Mathematics of the USSR-Sbornik, 1985, 51:2, 429–454

Bibliographic databases:

UDC: 511
MSC: Primary 10F37, 10F35; Secondary 13F20
Received: 20.04.1983

Citation: Yu. V. Nesterenko, “On algebraic independence of algebraic powers of algebraic numbers”, Mat. Sb. (N.S.), 123(165):4 (1984), 435–459; Math. USSR-Sb., 51:2 (1985), 429–454

Citation in format AMSBIB
\by Yu.~V.~Nesterenko
\paper On algebraic independence of algebraic powers of algebraic numbers
\jour Mat. Sb. (N.S.)
\yr 1984
\vol 123(165)
\issue 4
\pages 435--459
\jour Math. USSR-Sb.
\yr 1985
\vol 51
\issue 2
\pages 429--454

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    This publication is cited in the following articles:
    1. Yu. V. Nesterenko, “On the measure of algebraic independence of the values of an elliptic function at algebraic points”, Russian Math. Surveys, 40:4 (1985), 237–238  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. Yu. V. Nesterenko, “On a measure of the algebraic independence of the values of certain functions”, Math. USSR-Sb., 56:2 (1987), 545–567  mathnet  crossref  mathscinet  zmath  isi
    3. Nesterenko I., “The Measure of Algebraic Independence of Values of an Exponential Function”, 286, no. 4, 1986, 817–821  mathscinet  zmath  isi
    4. Nesterenko Y., “Measures of Algebraic Independence of Numbers and Functions”, Asterisque, 1987, no. 147-48, 141–149  mathscinet  zmath  isi
    5. Brownawell W., “Large Transcendence Degree Revisited .1. Exponential and Non-Cm Cases”, Lect. Notes Math., 1290 (1987), 149–173  crossref  mathscinet  zmath  isi
    6. Brownawell W. Tubbs R., “Large Transcendence Degree Revisited .2. the Cm Case”, Lect. Notes Math., 1290 (1987), 175–188  crossref  mathscinet  zmath  isi
    7. C.A.. Berenstein, D.C.. Struppa, “Small degree solutions for the polynomial Bezout equation”, Linear Algebra and its Applications, 98 (1988), 41  crossref  mathscinet  zmath
    8. Nesterenko Y., “Estimates of the Number of Null Functions of Certain Classes”, Acta Arith., 53:1 (1989), 29–46  crossref  mathscinet  zmath  isi
    9. Brownawell W., “Applications of Cayley-Chow Forms”, Lect. Notes Math., 1380 (1989), 1–78  crossref  mathscinet  isi
    10. Shestakova N., “Estimations of Mutual Transcendentality for Some Numbers”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1990, no. 6, 14–18  mathscinet  zmath  isi
    11. Amoroso F., “Polynomials with High Multiplicity”, Acta Arith., 56:4 (1990), 345–364  crossref  mathscinet  zmath  isi
    12. Shestakov S., “Transcendence Degree of Certain Fields Generated by Values of Elliptic Functions”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1991, no. 3, 30–36  mathscinet  isi
    13. Shestakov S., “On Some Numbers Algebraic Independence Measure”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1992, no. 2, 8–12  mathscinet  isi
    14. Ably M., “A Quantitative Version of the Lindemann-Weierstrass Theorem”, Acta Arith., 67:1 (1994), 29–45  crossref  mathscinet  zmath  isi
    15. Amoroso F., “Values of Polynomials with Integer Coefficients and Distance to their Common Zeros”, Acta Arith., 68:2 (1994), 101–112  crossref  mathscinet  zmath  isi
    16. Yu. V. Nesterenko, “On a measure of algebraic independence of values of an elliptic function”, Izv. Math., 59:4 (1995), 815–838  mathnet  crossref  mathscinet  zmath  isi
    17. Yu. V. Nesterenko, “Modular functions and transcendence questions”, Sb. Math., 187:9 (1996), 1319–1348  mathnet  crossref  crossref  mathscinet  zmath  isi
    18. Ya. M. Kholyavka, “On a measure of algebraic independence of values of Jacobi elliptic functions”, J. Math. Sci., 146:2 (2007), 5782–5790  mathnet  crossref  mathscinet  zmath  elib
    19. Roy D., “A Small Value Estimate for G(a) X G(M)”, Mathematika, 59:2 (2013), 333–363  crossref  mathscinet  zmath  isi
    20. Chirskii V.G., “Topical Problems of the Theory of Transcendental Numbers: Development of Approaches to Their Solution in the Works of Yu. V. Nesterenko”, Russ. J. Math. Phys., 24:2 (2017), 153–171  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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