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Independance algebrique de nombres de Liouville

  • Michel Waldschmidt
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1415)

Résumé

Nous donnons un aperçu historique sur le sujet, en distinguant deux types de résultats : d'une part ceux qui conduisent à produire des ensembles de nombres algébriquement indépendants (dont certains ont la puissance du continu) par des valeurs de séries lacunaires, d'autre part ceux qui reposent sur des énoncés d'approximation diophantienne, en particulier des mesures de transcendance.

Keywords

Algebraic Number Diophantine Approximation Arithmetic Property Algebraic Independence Transcendental Number 
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References

1. Construction de nombres algébriquement indépendants

  1. J. von NEUMANN.-Ein System algebraisch unabhängiger Zahlen; Math. Ann., 99 (1928), 134–141.MathSciNetCrossRefGoogle Scholar
  2. O. PERRON.-Uber mehrfach transzendente Erweiterungen des natürlichen Rationalitätsbereiches; Sitz. Bayer. Akad. Wiss., H2 (1932), 79–86.zbMATHGoogle Scholar
  3. H. KNESER.-Eine kontinuumsmächtige algebraisch unabhängige Menge reeller Zahlen; Bull. Soc. Math. Belg., 12 (1960), 23–27.MathSciNetzbMATHGoogle Scholar
  4. W.M. SCHMIDT.-Simultaneous approximation and algebraic independence of numbers; Bull. Amer. Math. Soc., 68 (1962), 475–478, et 69 (1963), 255.MathSciNetCrossRefzbMATHGoogle Scholar
  5. F. KUIPER and J. POPKEN.-On the so-called von Neumann numbers; Nederl. Akad. Wet. Proc. Ser. A, 65 (=Indag. Math., 24), (1962), 385–390.MathSciNetzbMATHGoogle Scholar
  6. A. DURAND.-Un système de nombres algébriquement indépendants; C.R. Acad. Sci. Paris Sér.A, 280 (1975), 309–311.MathSciNetzbMATHGoogle Scholar
  7. P. BUNDSCHUH und R. WALLISSER.-Algebraische Unabhängigkeit p-adischer Zahlen; Math. Ann., 221 (1976), 243–249.MathSciNetCrossRefzbMATHGoogle Scholar
  8. P. BUNDSCHUH.-Fractions continues et indépendance algébrique en p-adique; Journées Arith. Caen, Soc. Math. France Astérisque, 41–42 (1977), 179–181.MathSciNetzbMATHGoogle Scholar
  9. A. DURAND.-Indépendance algébrique de nombres complexes et critère de transcendance; Compositio Math., 35 (1977), 259–267.MathSciNetzbMATHGoogle Scholar
  10. W.W. ADAMS.-On the algebraic independence of certain Liouville numbers; J. Pure Appl. Algebra, 13 (1978), 41–47.MathSciNetCrossRefzbMATHGoogle Scholar
  11. P. BUNDSCHUH und F.J. WYLEGALA.-Uber algebraische Unabhängigkeit bei gewissen nichtfortsetzbaren Potenzreihen; Arch. Math., 34 (1980), 32–36.MathSciNetCrossRefzbMATHGoogle Scholar
  12. F.J. WYLEGALA.-Approximationsmaße und spezielle Systeme algebraische unabhängiger p-adischer Zahlen; Diss. Köln, 1980.Google Scholar
  13. I. SHIOKAWA.-Algebraic independence of certain gap series; Arch. Math., 38 (1982), 438–442.MathSciNetCrossRefzbMATHGoogle Scholar
  14. ZHU YAO CHEN.-Algebraic independence of the values of certain gap series in rational points; Acta Math. Sinica. 25 (1982), 333–339.MathSciNetzbMATHGoogle Scholar
  15. ZHU YAO CHEN.-On the algebraic independence of certain power series of algebraic numbers; Chin. Ann. of Math., 5B (1), (1984), 109–117.MathSciNetzbMATHGoogle Scholar
  16. NISHIOKA, K.-Algebraic independence of certain power series of algebraic numbers; J. Number Theory, 23 (1986), 354–364.MathSciNetCrossRefzbMATHGoogle Scholar
  17. NISHIOKA, K.-Algebraic independence of three Liouville numbers; Arch. Math., 47 (1986), 117–120.MathSciNetCrossRefzbMATHGoogle Scholar
  18. NISHIOKA, K.-Proof of Masser's conjecture on the algebraic independence of values of Liouville series; Proc. Japan Acad., Sér. A, 62 (1986), 219–222.MathSciNetCrossRefzbMATHGoogle Scholar
  19. NISHIOKA, K.-Conditions for algebraic independence of certain power series of algebraic numbers; Compositio Math., 62 (1987), 53–61.MathSciNetzbMATHGoogle Scholar
  20. ZHU YAO CHEN.-Arithmetic properties of gap series with algebraic coefficients; Acta Arith., 50 (1988). 295–308.MathSciNetzbMATHGoogle Scholar
  21. XU GUANG SHAN.-Diophantine approximation and transcendental number theory; in Number theory and its applications in China, Contemporary Math., 77 (1988), 127–142.MathSciNetCrossRefGoogle Scholar

2. Utilisation de résultats d'approximation

  1. D.D. MORDUHAT-BOLTOVSKOT.-Quelques propriétés des nombres transcendants de la première classe; [en russe, suivi d'un résumé en français] Mat. Sbornik, 34 (1927), 55–100.Google Scholar
  2. K. MAHLER.-Uber Beziehungen zwischen der Zahl e und den Liouvilleschen Zahlen; Math. Z., 31 (1930), 729–732.MathSciNetCrossRefzbMATHGoogle Scholar
  3. K. MAHLER.-Zur Approximation der Exponentialfunktion und des Logarithmus; J. reine angew. Math. (Crelle), 166 (1932), 118–150.MathSciNetzbMATHGoogle Scholar
  4. D.D. MORDOUKHAY-BOLTOVSKOY.-Sur les conditions pour qu'un nombre s'exprime au moyen d'équations transcendantes d'un type général; Dokl. Akad. Nauk. S.S.S.R., 52 (1946), 483–486 [Voir M.R. 8, 317g].MathSciNetzbMATHGoogle Scholar
  5. A.O. GEL'FOND.-The approximation of algebraic numbers by algebraic numbers and the theory of transcendental numbers; Usp. Mat. Nauk., 4 (32), (1949), 19–49. Trad. angl.: Amer. Math. Soc. Transl., 65 (1952), 81–124.MathSciNetGoogle Scholar
  6. Th. SCHNEIDER.-Introduction aux nombres transcendants; Springer, 1957; Trad. Franç. Gauthier Villars, 1959.Google Scholar
  7. N.I. FEL'DMAN.-Arithmetic properties of the solutions of a transcendental equation; Vestn. Mosk. Univ. Ser. I, Mat. Mec., fasc. 1 (1964), 13–20. Trad. angl.: Amer. Math. Soc. Transl., (2) 66 (1968), 145–153.MathSciNetGoogle Scholar
  8. A.A. SMELEV.-A.O. Gel'fond's method in the theory of transcendental numbers; Mat. Zam., 10 (1971), 415–426. Trad. angl.: Math. Notes, 10 (1971), 672–678.MathSciNetGoogle Scholar
  9. A.A. SMELEV.-On approximating the roots of some transcendental equations; Mat. Zam., 7 (1970), 203–210. Trad. angl.: Math. Notes, 7 (1970), 122–126.MathSciNetGoogle Scholar
  10. A.I. GALOCHKIN.-On diophantine approximations of values of an exponential function and solutions of some transcendental equations; Vestn. Mosk. Univ. Ser. I, Mat. Mec., fasc.3 (1972), 16–23.Google Scholar
  11. M. WALDSCHMIDT.-Approximation par des nombres algébriques des zéros de séries entières à coefficients algébriques; C.R. Acad. Sci. Paris Sér. A, 279 (1974), 793–796.MathSciNetzbMATHGoogle Scholar
  12. W.D. BROWNAWELL and M. WALDSCHMIDT.-The algebraic independence of certain numbers to algebraic powers; Acta Arith., 32 (1977), 63–71.MathSciNetzbMATHGoogle Scholar
  13. W.D. BROWNAWELL.-Algebraic independence of cubic powers of certain Liouville numbers; manuscrit, 1976.Google Scholar
  14. K. VÄÄNÄNEN.-On the arithmetic properties of certain values of the exponential function; Studia Sci. Math. Hungar., 11 (1976), 399–405.MathSciNetzbMATHGoogle Scholar
  15. K. VÄÄNÄNEN.-On the simultaneous approximation of certain numbers; J. reine angew. Math. (Crelle), 296 (1977), 205–211.MathSciNetzbMATHGoogle Scholar
  16. M. LAURENT.-Indépendance algébrique de nombres de Liouville à des puissances algébriques; Thèse 3ème cycle, Univ. Paris VI, Oct. 1977.Google Scholar
  17. M. LAURENT.-Indépendance algébrique de nombres de Liouville élevés à des puissances algébriques; C.R. Acad. Sci. Paris Sér. A, 286 (1978), 131–133.MathSciNetzbMATHGoogle Scholar
  18. M. WALDSCHMIDT.-Simultaneous approximation of numbers connected with the exponential function; J. Austral Math. Soc., 25 (1978), 466–478.MathSciNetCrossRefzbMATHGoogle Scholar
  19. A. BIJLSMA.-Simultaneous approximations in transcendental number theory; Acad. Proef., Amsterdam, 1978.Google Scholar
  20. F.J. WYLEGALA.-Approximationsmaße und spezielle Systeme algebraisch unabhängiger p-adischer Zahlen; Diss. Köln, 1980.Google Scholar
  21. A. FAISANT et G. PHILIBERT.-Quelques résultats de transcendance liés à l'invariant modulaire j; J. Number Theory, 25 (1987), 184–200.MathSciNetCrossRefzbMATHGoogle Scholar
  22. R. TUBBS.-A note on some elementary measures of algebraic independence; Proc. Amer. Math. Soc., à paraître.Google Scholar
  23. M. WALDSCHMIDT and ZHU YAOCHEN.-Algebraic independence of certain numbers related to Liouville numbers; Scientia Sinica Ser. A, à paraître.Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Michel Waldschmidt
    • 1
  1. 1.Institut Henri PoincaréC.N.R.S. U.A. 763 (Problèmes Diophantiens)Paris Cedex 05

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