Démonstration du théorème de Baker-Feldman via les formes linéaires en deux logarithmes

Yuri Bilu; Yann Bugeaud

Journal de théorie des nombres de Bordeaux (2000)

  • Volume: 12, Issue: 1, page 13-23
  • ISSN: 1246-7405

Abstract

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We show that the Liouville-Baker-Feldman inequality | α - y / x | ≫ eff x γ - n easily follows from an estimate for linear forms in two logarithms.

How to cite

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Bilu, Yuri, and Bugeaud, Yann. "Démonstration du théorème de Baker-Feldman via les formes linéaires en deux logarithmes." Journal de théorie des nombres de Bordeaux 12.1 (2000): 13-23. <http://eudml.org/doc/248497>.

@article{Bilu2000,
abstract = {Nous montrons que l’inégalité de Liouville-Baker-Feldman $| \alpha - y/x| \gg _\{\mathrm \{eff\}\} x^\{\gamma -n\}$ est une conséquence facile d’une minoration de formes linéaires en deux logarithmes.},
author = {Bilu, Yuri, Bugeaud, Yann},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {measure of linear independence; logarithms of algebraic numbers; measures of linear independence for two logarithms; Schneider's transcendence method},
language = {fre},
number = {1},
pages = {13-23},
publisher = {Université Bordeaux I},
title = {Démonstration du théorème de Baker-Feldman via les formes linéaires en deux logarithmes},
url = {http://eudml.org/doc/248497},
volume = {12},
year = {2000},
}

TY - JOUR
AU - Bilu, Yuri
AU - Bugeaud, Yann
TI - Démonstration du théorème de Baker-Feldman via les formes linéaires en deux logarithmes
JO - Journal de théorie des nombres de Bordeaux
PY - 2000
PB - Université Bordeaux I
VL - 12
IS - 1
SP - 13
EP - 23
AB - Nous montrons que l’inégalité de Liouville-Baker-Feldman $| \alpha - y/x| \gg _{\mathrm {eff}} x^{\gamma -n}$ est une conséquence facile d’une minoration de formes linéaires en deux logarithmes.
LA - fre
KW - measure of linear independence; logarithms of algebraic numbers; measures of linear independence for two logarithms; Schneider's transcendence method
UR - http://eudml.org/doc/248497
ER -

References

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  1. [1] A. Baker, Linear forms in the logarithms of algebraic numbers I-IV. Mathematika13 (1966), 204-216; 14 (1967), 102-107 et 220-224; 15 (1968), 204-216. Zbl0161.05201MR258756
  2. [2] A. Baker, Contributions to the theory of Diophantine equations. I. On the representation of integers by binary forms. Phil. Trans. R. Soc. London Ser.A263 (1967 -68), 173-191. Zbl0157.09702MR228424
  3. [3] A. Baker, A sharpening of the bounds for linear forms in logarithms I-III. Acta Arith.21 (1972), 117-129; 24 (1973), 33-36; 27 (1975), 247-252. Zbl0301.10030
  4. [4] A. Baker, G. Wüstholz, Logarithmic forms and group varieties. J. Reine Angew. Math.442 (1993), 19-62. Zbl0788.11026MR1234835
  5. [5] E. Bombieri, Effective Diophantine Approximation on Gm. Ann. Scuola Norm. Sup. Pisa Cl. Sci.20 (1993), 61-89. Zbl0774.11034MR1215999
  6. [6] E. Bombieri, P.B. Cohen, Effective Diophantine Approximation on (Gm, II. Ann. Scuola Norm. Sup. Pisa Cl. Sci.24 (1997), 205-225. Zbl0912.11028MR1487954
  7. [7] Y. Bugeaud, Bornes effectives pour les solutions des équations en S-unités et des équations de Thue-Mahler. J. Number Theory71 (1998), 227-244. Zbl0926.11015MR1633809
  8. [8] Y. Bugeaud, K. Gyôry, Bounds for the solutions of Thue-Mahler equations and norm form equations. Acta Arith.74 (1996), 273-292. Zbl0861.11024MR1373714
  9. [9] N.I. Feldman, Improved estimate for a linear form of the logarithms of algebraic numbers, (en russe). Mat. Sb.77 (1968), 256-270. Également: Math. USSR. Sb.6 (1968) 393-406. Zbl0235.10018MR232736
  10. [10] N.I. Feldman, An effective refinement of the exponent in Liouville's theorem, (en russe). Iz. Akad. Nauk SSSR, Ser. Mat.35 (1971), 973-990. Également: Math. USSR. Izv.5 (1971) 985-1002. Zbl0259.10031MR289418
  11. [11] M. Laurent, Linear forms in two logarithms and interpolation determinants. Acta Arith.66 (1994), 181-199. Zbl0801.11034MR1276987
  12. [12] M. Laurent, M. Mignotte, Y. Nesterenko, Formes linéaires en deux logarithmes et déterminants d'interpolation. J. Number Theory55 (1995), 285-321. Zbl0843.11036MR1366574
  13. [13] M. Waldschmidt, Minorations de combinaisons linéaires de logarithmes de nombres algébriques. Canadian J. Math.45 (1993), 176-224. Zbl0774.11036MR1200327

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