Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T09:02:49.806Z Has data issue: false hasContentIssue false
  • English
  • Français

Transcendence measures for exponentials and logarithms

Published online by Cambridge University Press:  09 April 2009

Michel Waldschmidt
Michel Waldschmidt
Affiliation:
Université P. et M. Curie(Paris VI) Mathématiques, T. 45-46 4, Place Jussieu 75230 Paris Cedex 05, France
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the present paper, we derive transcendence measures for the numbers log α, eβ, αβ, (log α1)/(log α2) from a previous lower bound of ours on linear forms in the logarithms of algebraic numbers.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 25 , Issue 4 , June 1978 , pp. 445 - 465
Copyright
Copyright © Australian Mathematical Society 1978

References

[B]Baker, A. (1975), Transcendental Number Theory (Cambridge University Press).CrossRefGoogle Scholar
[Ch]Cudnovskii, G. V. (1974), “The Gelfond-Baker method in problems of diophantine approximation”, Colloquia Math. Soc. János Bolyai 13, 19–30 (North Holland, Amsterdam, 1975).Google Scholar
[Ci 1]Cijsouw, P. L. (1972), Transcendence Measures (Acad. Proefschrift, Amsterdam).Google Scholar
[Ci 2]Cijsouw, P. L. (1974), “Transcendence measures of exponentials and logarithms of algebraic numbers”, Compositio Math. 28, 163–178.Google Scholar
[Ci 3]Cijsouw, P. L. (1974), “Transcendence measures of certain numbers whose transcendency was proved by A. Baker”, Compositio Math. 28, 179–194.Google Scholar
[Ci 4 ]Cijsouw, P. L. (1974/1975), “On the approximability of the logarithms of algebraic numbers”, Sém. DELANGE-PISOT-POITOU (Théorie des Nombres), 16e année no. 19.Google Scholar
[Ci 5]Cijsouw, P. L. (1977), “A transcendence measure for π”, Transcendence Theory, Advances and Applications, edited by Baker, A. and Masser, D. W. (Academic Press, New York).Google Scholar
[C]Cijsouw, P. L. and Waldschmidt, M. (1977), “Linear forms and simultaneous approximations”, Compositio Math. 34, 173–197.Google Scholar
[F 1]Fel'dman, N. I. (1966), “Approximation of certain transcendental numbers. I. The approximation of logarithms of algebraic numbers”, English transl., Amer. Math. Soc. Transl. II ser., 59, 224–245.Google Scholar
[F 2]Fel'dman, N. I. (1966), “On the measure of transcendence of π”, English transl., Amer. Math. Soc. Transl. II ser., 58, 110–124.Google Scholar
[F 3]Fel'dman, (1966), “Approximation of the logarithms of algebraic numbers by algebraic numbers”, English transl., Amer. Math. Soc. Transl. II ser., 58, 125–142.Google Scholar
[F 4]Fel'dman, N. I. (1977), “Approximation of number π by algebraic numbers from special fields”, J. Number Theory 9, 48–60.CrossRefGoogle Scholar
[F-S]Fel'dman, N. I. and Šidlovskii, A. B. (1967), “The development and present state of the theory of transcendental numbers”, English transl., Russian Math. Surveys, 22, 1–79.CrossRefGoogle Scholar
[G]Gel'fond, A. O. (1960), Transcendental and Algebraic Numbers, English transl. (Dover Publications, New York).Google Scholar
[K-P]Koksma, J. F. und Popken, J. (1932), “Zur Transzendenz von eπ”, J. reine angew. Math. 168, 211–230.Google Scholar
[Ma 1]Mahler, K. (1932), “Zur Approximation der Exponentialfunktion und des Logarithmus”, J. reine angew. Math. 166, 118–150.CrossRefGoogle Scholar
[Ma 2]Mahler, K. (1953), “On the approximation of logarithms of algebraic numbers”, Philos. Trans. Roy. Soc. London, Ser. A, 245, 371–398.Google Scholar
[Ma 3]Mahler, K. (1953), “On the approximation of π”, Nederl. Akad. Wetensch. Proc., Ser. A, 56 ( = Indag. Math. 15), 30–42.Google Scholar
[Ma 4]Mahler, (1962), “On some inqualities for polynomials in several variables”, J. London Math. Soc. 37, 341–344.CrossRefGoogle Scholar
[Ma 5]Mahler, K. (1967), “Application of some formulae by Hermite to the approximation of exponentials and logarithms”, Math. Ann. 168, 200–227.CrossRefGoogle Scholar
[Mi]Mignotte, M. (1974), “Approximations rationnelles de π et quelques autres nombres”, Journées Arithm. [1973, Grenoblel], Bull. Soc. Math. France, Mémoire 37, 121–132.Google Scholar
[M-W]Mignotte, M. and Waldschmidt, M. (1978), “Linear forms in two logarithms and Schneider's method”, Math. Annalen. 231, 241–267.CrossRefGoogle Scholar
[P]Popken, J. (1929) “Zur Transzendenz von e”, Math. Zeitscht. 29, 525–541.CrossRefGoogle Scholar
[W]Waldschmidt, M. (to appear), “A lower bound for linear forms in logarithms”, Acta Arith. 37 (1979).Google Scholar