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Effective diophantine approximation on $\mathbb{G}_M$, II

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  • Law
  • Mathematics

Abstract

Effective diophantine approximation on GM, II ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze E. BOMBIERI P. B. COHEN Effective diophantine approximation onGM, II Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, tome 24, no 2 (1997), p. 205-225. <http://www.numdam.org/item?id=ASNSP_1997_4_24_2_205_0> © Scuola Normale Superiore, Pisa, 1997, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Effective Diophantine Approximation on GM, II E. BOMBIERI - P. B. COHEN 1. - Introduction In [Bo], Theorem 1 a result on effective approximations at archimedean places to roots of high order of algebraic numbers was obtained and applied to give an effective result on archimedean diophantine approximation in a number field by a finitely generated multiplicative subgroup. The result obtained there was strong enough to derive a new effective solution of Thue’s equation and a new proof of the Baker-Feldman theorem on approximations of algebraic numbers by rationals. In this article we derive analogous results in the non-archimedean case. We obtain analogues of Theorems 1 and 2 in [Bo], incorporating in particular the refinements announced there. In fact, we are able to improve on the previous treatment in several respects, by introducing new tools and ideas. The new tool used is M. Laurent’s determinantal method, which replaces the traditional Siegel lemma arguments along the lines followed in recent work of P. Corvaja

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