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Diophantine Approximation pp 249–344Cite as

Linear Independence Measures for Logarithms of Algebraic Numbers

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Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1819))

Abstract

Let α1,...,α n be nonzero algebraic numbers and b 1,...,b n rational integers. Assume α b11 ...α bn n ≠ 1. According to Liouville’s inequality (Proposition 1.13), the lower bound

$$ \left| {\alpha _1^{b_1 } ...\alpha _n^{b_n } - 1} \right| \geqslant e^{ - cB} $$

holds with B=max{|b 1|,...,|b n |} and with a positive number c depending only on α1,...,α n . A fundamental problem is to prove a sharper estimate.

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References

  1. Baker, A. — Linear forms in the logarithms of algebraic numbers. I, II, III, IV. Mathematika 13 (1966), 204–216; ibid. 14 (1967), 102–107; ibid. 14 (1967), 220–228; ibid. 15 (1968), 204–216.

    Article  Google Scholar 

  2. Baker, A. — Transcendental number theory. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1975. Second edition, 1990.

    MATH  Google Scholar 

  3. Baker, A. — The theory of linear forms in logarithms. Transcendence theory: advances and applications (Proc. Conf., Univ. Cambridge, Cambridge, 1976), pp. 1–27. Academic Press, London, 1977.

    Google Scholar 

  4. Baker, A.; Stark, H. M. — On a fundamental inequality in number theory. Ann. of Math. (2) 94 (1971), 190–199.

    Article  MathSciNet  Google Scholar 

  5. Baker, A.; Wüstholz, G. — Logarithmic forms and group varieties. J. reine angew. Math. 442 (1993), 19–62.

    MATH  MathSciNet  Google Scholar 

  6. Fel’dman, N. I. — Hilbert’s seventh problem. Moskov. Gos. Univ., Moscow, 1982.

    Google Scholar 

  7. Fel’dman, N. I.; Nesterenko, Y. V. — Number theory. IV. Transcendental Numbers. Encyclopaedia of Mathematical Sciences 44. Springer-Verlag, Berlin, 1998.

    Google Scholar 

  8. Gaál, I.; Lettl, G. — A parametric family of quintic Thue equations II. Monatsh. Math. 131, No. 1 (2000), 29–35.

    Article  MATH  MathSciNet  Google Scholar 

  9. Gramain, F. — Lemme de Schwarz pour des produits cartésiens. Ann. Math. Blaise Pascal 8, No. 2 (2001), 67–75.

    MATH  MathSciNet  Google Scholar 

  10. Koksma, J. F. — Diophantische Approximationen. Reprint. Springer-Verlag, Berlin-New York, 1974.

    Google Scholar 

  11. Lang, S. — Elliptic curves: Diophantine analysis. Grundlehren der Mathematischen Wissenschaften 231. Springer-Verlag, Berlin-New York, 1978.

    MATH  Google Scholar 

  12. Lang, S. — Algebra. Third edition. Addison-Wesley Publishing Co., Reading, Mass., 1993.

    MATH  Google Scholar 

  13. Laurent, M. — Sur quelques résultats récents de transcendance. Journées Arithmétiques, 1989 (Luminy, 1989). Astérisque No. 198–200 (1991), 209–230.

    MathSciNet  Google Scholar 

  14. Laurent, M. — Linear forms in two logarithms and interpolation determinants. Acta Arith. 66, No. 2 (1994), 181–199.

    MATH  MathSciNet  Google Scholar 

  15. Laurent, M.; Roy, D. — Criteria of algebraic independence with multiplicities and approximation by hypersurfaces; J. reine angew. Math. 536 (2001), 65–114.

    MATH  MathSciNet  Google Scholar 

  16. Laurent, M.; Mignotte, M.; Nesterenko, Y. V. — Formes linéaires en deux logarithmes et déterminants d’interpolation. J. Number Theory 55, No. 2 (1995), 285–321.

    Article  MATH  MathSciNet  Google Scholar 

  17. Mahler, K. — On the approximation of logarithms of algebraic numbers. Philos. Trans. Roy. Soc. London. Ser. A. 245 (1953), 371–398.

    Article  MathSciNet  Google Scholar 

  18. Mahler, K. — Applications of some formulae by Hermite to the approximation of exponentials and logarithms. Math. Ann. 168 (1967), 200–227.

    Article  MATH  MathSciNet  Google Scholar 

  19. Masser, D.W. — Heights, transcendence, and linear independence on commutative group varieties. This volume.

    Google Scholar 

  20. Matveev, E. M. — An explicit lower bound for a homogeneous rational linear forms in the logarithms of algebraic numbers. I and II. Izv. Akad. Nauk SSSR. Ser. Mat. 62 No. 4 (1998), 81–136; ibid. 64 No. 6 (2000), 125–180; Engl. transl.: Izv. Math. 62 No. 4 (1998), 723–772; ibid. 64 No. 6 (2000), 1217–1269.

    Google Scholar 

  21. Nesterenko, Y. V.; Waldschmidt, M. — On the approximation of the values of exponential function and logarithm by algebraic numbers. (Russian) Diophantine approximations, Proceedings of papers dedicated to the memory of Prof. N. I. Fel’dman, ed. Yu. V. Nesterenko, Centre for applied research under Mech.-Math. Faculty of MSU, Moscow (1996), 23–42. Number Theory xxx Math Archives: http://arXiv.org/abs/math/0002047

    Google Scholar 

  22. Philippon, P. — Nouveaux lemmes de zéeros dans les groupes algébriques commutatifs. Symposium on Diophantine Problems (Boulder, CO, 1994). Rocky Mountain J. Math. 26 No. 3 (1996), 1069–1088.

    Article  MATH  MathSciNet  Google Scholar 

  23. Philippon, P. — Une approche méthodique pour la transcendance et l’indépendance algébrique de valeurs de fonctions analytiques. J. Number Theory 64 No. 2 (1997), 291–338.

    Article  MATH  MathSciNet  Google Scholar 

  24. Philippon, P. and Waldschmidt, M. — Lower bounds for linear forms in logarithms. New advances in transcendence theory (Durham, 1986), 280–312, Cambridge Univ. Press, Cambridge-New York, 1988.

    Google Scholar 

  25. Philippon, P. and Waldschmidt, M. — Formes linéaires de logarithmes sur les groupes algébriques commutatifs. Illinois J. Math. 32 No. 2 (1988), 281–314.

    MATH  MathSciNet  Google Scholar 

  26. Roy, D. — Interpolation formulas and auxiliary functions. J. Number Theory 94 No. 2 (2002), 248–285.

    Article  MATH  MathSciNet  Google Scholar 

  27. Schmidt, W. M.— Diophantine approximation. Lecture Notes in Mathematics, 785. Springer, Berlin, 1980.

    MATH  Google Scholar 

  28. Sprindžuk, V. G. — Classical Diophantine equations. Translated from the 1982 Russian original (Nauka, Moscow). Translation edited by Ross Talent and Alf van der Poorten. Lecture Notes in Mathematics 1559. Springer-Verlag, Berlin, 1993.

    Google Scholar 

  29. Tijdeman, R. — On the number of zeros of general exponential polynomials. Nederl. Akad. Wetensch. Proc. Ser. A 74 = Indag. Math. 33 (1971), 1–7.

    MathSciNet  Google Scholar 

  30. Waldschmidt, M. — Nombres transcendants. Lecture Notes in Mathematics 402. Springer-Verlag, Berlin-New York, 1974.

    MATH  Google Scholar 

  31. Waldschmidt, M. — A lower bound for linear forms in logarithms. Acta Arith. 37 (1980), 257–283.

    MATH  MathSciNet  Google Scholar 

  32. Waldschmidt, M. — Linear independence of logarithms of algebraic numbers. The Institute of Mathematical Sciences, Madras, IMSc Report No. 116 (1992), 168 pp. http://www.math.jussieu.fr/~miw/articles/IMSc.Rpt.116.html

    MATH  Google Scholar 

  33. Waldschmidt, M. — Integer valued functions on products. J. Ramanujan Math. Soc. 12 No. 1 (1997), 1–24.

    MATH  MathSciNet  Google Scholar 

  34. Waldschmidt, M. — Diophantine Approximation on Linear Algebraic Groups. Transcendence Properties of the Exponential Function in Several Variables. Grundlehren der Mathematischen Wissenschaften 326, Springer-Verlag, Berlin-Heidelberg, 2000.

    MATH  Google Scholar 

  35. Waldschmidt, M. — On a Problem of Mahler Concerning the Approximation of Exponentials and Logarithms. Publ. Math. Debrecen 56 No. 3–4 (2000), 713–738.

    MATH  MathSciNet  Google Scholar 

  36. Wielonsky, F. — Hermite-Padé approximants to exponential functions and an inequality of Mahler. J. Number Theory 74 No. 2 (1999), 230–249.

    Article  MATH  MathSciNet  Google Scholar 

  37. Yu, Kun Rui — Linear forms in p-adic logarithms. Acta Arith. 53 (1989), No. 2, 107–186. II. Compositio Math. 74, No. 1 (1990), 15–113 and 76 (1990), No. 1–2, 307. III. Compositio Math. 91, No. 3 (1994), 241–276.

    MATH  MathSciNet  Google Scholar 

  38. Yu, Kun Rui — p-adic logarithmic forms and Group Varieties I. J. reine angew. Math. 502 (1998), 29–92. II. Acta Arith. 89 No. 4 (1999), 337–378.

    MATH  MathSciNet  Google Scholar 

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  1. Institut de Mathématiques, Université Pierre et Marie Curie (Paris VI), Paris

    Michel Waldschmidt

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  1. Michel Waldschmidt
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  1. Laboratoire de Mathématiques Nicolas Oresme, CNRS UMR 6139, Université de Caen, BP 5186, 14032, Caen, France

    Francesco Amoroso

  2. Instituto Universitario Architettura-D.C.A., Santa Croce 191, 300135, Venezia, Italy

    Umberto Zannier

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Waldschmidt, M. (2003). Linear Independence Measures for Logarithms of Algebraic Numbers. In: Amoroso, F., Zannier, U. (eds) Diophantine Approximation. Lecture Notes in Mathematics, vol 1819. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44979-5_5

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  • DOI: https://doi.org/10.1007/3-540-44979-5_5

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