Hostname: page-component-848d4c4894-sjtt6 Total loading time: 0 Render date: 2024-07-07T18:12:24.706Z Has data issue: false hasContentIssue false
  • English
  • Français

Quelques remarques sur des questions d'approximation diophantienne

Published online by Cambridge University Press:  17 April 2009

Patrice Philippon
Patrice Philippon
Affiliation:
UMR 9994 du CNRS-Problèmes Diophantiens, Université P. & M. Curie, T46-56, 5ème ét., F-75252 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.

Hoping for a hand-shake between methods from diophantine approximation theory and transcendance theory, we show how zeros estimates from transcendance theory imply Roth's type lemmas (including the product theorem). We also formulate some strong conjectures on lower bounds for linear forms in logarithms of rational numbers with rational coefficients, inspired by the subspace theorem and which would imply, for example, the abc conjecture.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 59 , Issue 2 , April 1999 , pp. 323 - 334
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Baker, A., ‘Linear forms in logarithms and abc-conjecture’, in Proc. Conference in Eger (W. de Gruyter, Berlin, 1998).Google Scholar
[2]Evertse, J.H., ‘An explicit version of Falting's product theorem and an improvement of Roth's lemma’, Acta Arith. 73 (1995), 215–248.CrossRefGoogle Scholar
[3]Faltings, G., ‘Diophantine approximations on abelian varieties’, Ann. of Math. 133 (1991), 549–576.CrossRefGoogle Scholar
[4]Ferretti, R., ‘An effective version of Falting's product theorem’, Forum Math. 8 (1996), 401–427.CrossRefGoogle Scholar
[5]Lang, S., Elliptic curves: diophantine analysis (Springer Verlag, Berlin, Heidelberg, New York, 1978).CrossRefGoogle Scholar
[6]Nakamaye, M., ‘Multiplicity estimates and the product theorem’, Bull. Soc. Math. France 123 (1995), 155–188.Google Scholar
[7]Philippon, P., ‘Nouveaux lemmes de zéros dans les groupes algébriques commutatifs’, Rocky Mountain J. Math. 26 (1996), 1069–1088.CrossRefGoogle Scholar
[8]Schlickewei, H.P., ‘The p-adic Thue-Siegel-Roth-Schmidt theorem’, Arch. Math. 29 (1977), 267–270.CrossRefGoogle Scholar
[9]Schmidt, W.M., ‘The subspace theorem in diophantine approximation’, Compositio Math. 69 (1989), 121–173.Google Scholar
[10]Vojta, P., Diophantine approximations and value distribution theory, Lecture Notes in Math. 1239 (Springer Verlag, Berlin, Heidelberg, New York, 1997).Google Scholar