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Large transcendence degree revisited

I. Exponential and non-CM cases

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Diophantine Approximation and Transcendence Theory

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Bibliography

  1. Brownawell, W.D. and Masser, D.W. Multiplicity estimates for analytic functions II, Duke Math J. 47(1980), 273–295.

    Article  MathSciNet  MATH  Google Scholar 

  2. Brownawell, W.D. On the development of Gelfond's method, pp. 16–44 in Proc. Number Theory Carbondale 1979, M.V. Nathanson, ed., Lecture Notes in Math. 751, Springer Verlag, Berlin-Heidelberg-New York, 1979.

    Google Scholar 

  3. Brownawell, W.D. Bounds on the degree in the Nullstellensatz, Annals of Math., to appear.

    Google Scholar 

  4. Brownawell, W.D. A local Diophantine Nullstellen inequality, manuscript.

    Google Scholar 

  5. Chudnovsky, G.V. Some analytic methods in the theory of transcendental numbers, Inst. of Math., Ukr, SSR Acad. Sci., Preprint IM 74-8 and IM 74-9, Kiev, 1974 = Chapter 1 in Contributions to the Theory of Transcendental Numbers, Am. Math. Soc., Providence, R.I. 1984.

    Chapter  Google Scholar 

  6. Endell, R. Zur algebraischen Unabhängigkeit gewisser Werte der Exponentialfunktion, in Number Theory Noordwijkerhout 1983, H. Jager, ed., Lecture Notes in Math. 1068, Springer Verlag, Berlin-Heidelberg-New York, 1984.

    Chapter  Google Scholar 

  7. Gelfond, A.O. Transcendental and Algebraic Numbers, GITTL Moscow 1952 = Dover, New York, 1960.

    Google Scholar 

  8. Jabbouri, E.M. Mesures d'indépendance algébriques sur les groupes algébriques commutatifs, manuscript.

    Google Scholar 

  9. Masser, D.W. On polynomials and exponential polynomials in several complex variables, Invent. Math. 63(1981), 81–95.

    Article  MathSciNet  MATH  Google Scholar 

  10. Masser, D.W. and Wüstholz, G. Zero estimates on group varieties I, Invent. Math. 64(1981), 489–516.

    Article  MathSciNet  MATH  Google Scholar 

  11. Masser, D.W. and Wüstholz, G. Fields of large transcendence degree generated by values of elliptic functions, Invent. Math. 72(1983), 407–463.

    Article  MathSciNet  MATH  Google Scholar 

  12. Nesterenko, Yu.V. Estimates for the orders of zeros of functions of a certain class and applications in the theory of transcendental numbers, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), 253–284 = Math. USSR Izv. 11(1977), 239–270.

    MathSciNet  MATH  Google Scholar 

  13. Nesterenko, Yu.V. Bounds for the characteristic function of a prime ideal, Mat. Sbornik 123, No. 1(1984), 11–34 = Math. USSR Sbornik 51(1985), 9–32.

    MathSciNet  Google Scholar 

  14. Nesterenko, Yu.V. On algebraic independence of algebraic powers of algebraic numbers, Mat. Sbornik 123, No. 4(1984), 435–459 = Math. USSR Sbornik 51(1985), 429–454, brief version in Approximations Diophantiennes et Nombres Transcendants, D. Bertrand and M. Waldschmidt, eds, Birkhäuser Verlag, Verlag, Boston-Basel-Stuttgart, 1983, pp.199–220.

    MathSciNet  MATH  Google Scholar 

  15. Nesterenko, Yu.V. On a measure of the algebraic independence of the values of some functions, Mat. Sbornik 128, No. 4(1985), 545–568 = Math USSR Sbornik 56(1986), in press.

    MathSciNet  MATH  Google Scholar 

  16. Nesterenko, Yu.V. On bounds of measures of algebraically independent numbers, pp.65–76 in Diophantine Approximations, P.L. Ulnova, ed., Moscow Univ. Press, Moscow, 1985 (Russian).

    Google Scholar 

  17. Philippon, P. Indépendance algébrique de valeurs des fonctions exponentielles p-adiques, J. reine angew. Math. 329 (1981), 42–51.

    MathSciNet  MATH  Google Scholar 

  18. Philippon, P. Pour une théorie de l'indépendance algébrique, Thèse, Université de Paris XI, 1983.

    Google Scholar 

  19. Philippon, P. Sur les mesures d'indépendance algébrique, pp.219–233 in Seminaire de Theorie des Nombres, Catherine Goldstein, ed, Birkhäuser, Boston-Basel-Stuttgart, 1985.

    Google Scholar 

  20. Philippon, P. Critères pour l'indépendance algébrique, Inst. Hautes Etudes Sci. Publ. Math. No. 64, 1986, 5–52.

    Google Scholar 

  21. Philippon, P. Lemmes de zéros dans les groupes algébriques commutatifs, Bull. Soc. Math. France.114(1986), 355–383.

    MathSciNet  MATH  Google Scholar 

  22. Reyssat, E. Un critère d'indépendance algébrique, J. reine angew. Math. 329(1981), 66–81.

    MathSciNet  MATH  Google Scholar 

  23. Samuel, P. Méthodes d'Algèbre Abstraite en Géometrie Algébrique, 2nd ed., Ergebnisse d. Math. u. ihrer Grenzgebiete, Band 4, Springer Verlag, Berlin-Heidelberg-New York, 1967.

    Google Scholar 

  24. Tijdeman, R. An auxiliary result in the theory of transcendental numbers, J. Number Theory 5(1973), 80–94.

    Article  MathSciNet  MATH  Google Scholar 

  25. Tubbs, R. Lower bounds on linear forms and algebraic independence, Special Session on Transcendence Theory and Diophantine Problems, New York Regional AMS Meeting, April 15, 1983.

    Google Scholar 

  26. Van der Waerden, B.L. Zur algebraischen Geometrie 19, Grundpolynom und zugeordnete Form, Math. Ann. 136(1958), 139–155.

    Article  MathSciNet  MATH  Google Scholar 

  27. Waldschmidt, M. Nombres Transcendants et Groupes Algébriques, Astérisque 69–70, Soc. Math. France, Paris, 1979.

    MATH  Google Scholar 

  28. Waldschmidt, M. Algebraic independence of transcendental numbers. Gel'fond's method and its developments, pp.551–571, in Perspectives in Mathematics, Anniversary of Oberwolfach, W. Jager, J. Moser, R. Remmert, eds, Birkhäuser Verlag, Boston-Basel-Stuttgart, 1984.

    Google Scholar 

  29. Waldschmidt, M. Groupes algébriques et grands degrés de transcendance, Acta Math. 156(1986), 253–302.

    Article  MathSciNet  MATH  Google Scholar 

  30. Waldschmidt, M. Algebraic independence of values of exponential and elliptic functions, J. Indian Math. Soc. 48 (1984), 215–228.

    MathSciNet  MATH  Google Scholar 

  31. Waldschmidt, M. and Zhu Yao Chen. Une généralization en plusieurs variables d'un critère de transcendance de Gel'fond, C.R. Acad. Sc. Paris, Ser. I, 297(1983), 229–232.

    MathSciNet  Google Scholar 

  32. Warkentin, P. Algebraische Unabhängigkeit gewisser p-adischer Zahlen, Diplomarbeit, Freiburg, 1978.

    Google Scholar 

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Gisbert Wüstholz

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© 1987 Springer-Verlag

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Brownawell, W.D. (1987). Large transcendence degree revisited. In: Wüstholz, G. (eds) Diophantine Approximation and Transcendence Theory. Lecture Notes in Mathematics, vol 1290. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0078707

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  • DOI: https://doi.org/10.1007/BFb0078707

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