Explicit lower bounds for linear forms in two logarithms

Nicolas Gouillon[1]

  • [1] Institut de Mathématiques de Luminy 163, Avenue de Luminy, case 907 13288 Marseille Cedex 9, France

Journal de Théorie des Nombres de Bordeaux (2006)

  • Volume: 18, Issue: 1, page 125-146
  • ISSN: 1246-7405

Abstract

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We give an explicit lower bound for linear forms in two logarithms. For this we specialize the so-called Schneider method with multiplicity described in [10]. We substantially improve the numerical constants involved in existing statements for linear forms in two logarithms, obtained from Baker’s method or Schneider’s method with multiplicity. Our constant is around 5 . 10 4 instead of 10 8 .

How to cite

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Gouillon, Nicolas. "Explicit lower bounds for linear forms in two logarithms." Journal de Théorie des Nombres de Bordeaux 18.1 (2006): 125-146. <http://eudml.org/doc/249609>.

@article{Gouillon2006,
abstract = {We give an explicit lower bound for linear forms in two logarithms. For this we specialize the so-called Schneider method with multiplicity described in [10]. We substantially improve the numerical constants involved in existing statements for linear forms in two logarithms, obtained from Baker’s method or Schneider’s method with multiplicity. Our constant is around $5.10^\{4\}$ instead of $10^\{8\}$.},
affiliation = {Institut de Mathématiques de Luminy 163, Avenue de Luminy, case 907 13288 Marseille Cedex 9, France},
author = {Gouillon, Nicolas},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {linear form in the logarithms of algebraic numbers},
language = {eng},
number = {1},
pages = {125-146},
publisher = {Université Bordeaux 1},
title = {Explicit lower bounds for linear forms in two logarithms},
url = {http://eudml.org/doc/249609},
volume = {18},
year = {2006},
}

TY - JOUR
AU - Gouillon, Nicolas
TI - Explicit lower bounds for linear forms in two logarithms
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 1
SP - 125
EP - 146
AB - We give an explicit lower bound for linear forms in two logarithms. For this we specialize the so-called Schneider method with multiplicity described in [10]. We substantially improve the numerical constants involved in existing statements for linear forms in two logarithms, obtained from Baker’s method or Schneider’s method with multiplicity. Our constant is around $5.10^{4}$ instead of $10^{8}$.
LA - eng
KW - linear form in the logarithms of algebraic numbers
UR - http://eudml.org/doc/249609
ER -

References

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  1. Alan Baker, Transcendental number theory. Cambridge University Press, London, 1975. Zbl0297.10013MR422171
  2. Nicolas Gouillon, Un lemme de zéros. C. R. Math. Acad. Sci. Paris, 335 (2) (2002), 167–170. Zbl1031.11046MR1920014
  3. Nicolas Gouillon, Minorations explicites de formes linéaires en deux logarithmes. Thesis (2003) : http://tel.ccsd.cnrs.fr/documents/archives0/00/00/39/64/index_fr.html 
  4. Michel Laurent, Linear forms in two logarithms and interpolation determinants. Acta Arith. 66 (2) (1994), 181–199. Zbl0801.11034MR1276987
  5. Michel Laurent, Maurice Mignotte, Yuri Nesterenko, Formes linéaires en deux logarithmes et déterminants d’interpolation. J. Number Theory 55 (2) (1995), 285–321. Zbl0843.11036MR1366574
  6. D. W. Masser, On polynomials and exponential polynomials in several complex variables. Invent. Math. 63 (1) (1981), 81–95. Zbl0436.32005MR608529
  7. E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II. Izv. Ross. Akad. Nauk Ser. Mat. 64 (6) (2000), 125–180. Zbl1013.11043MR1817252
  8. Maurice Mignotte, Michel Waldschmidt, Linear forms in two logarithms and Schneider’s method. II. Acta Arith. 53 (3) (1989), 251–287. Zbl0642.10034MR1032827
  9. Michel Waldschmidt, Minorations de combinaisons linéaires de logarithmes de nombres algébriques. Canad. J. Math. 45 (1) (1993), 176–224. Zbl0774.11036MR1200327
  10. Michel Waldschmidt, Diophantine Approximation on Linear Algebraic Groups. Springer-Verlag, 1999. Zbl0944.11024MR1756786
  11. Kun Rui Yu, Linear forms in p -adic logarithms. Acta Arith. 53 (2) (1989), 107–186. Zbl0699.10050MR1027200

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