Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T02:22:19.321Z Has data issue: false hasContentIssue false
  • English
  • Français

Classification de Mahler et distances locales

Published online by Cambridge University Press:  17 April 2009

Patrice Philippon
Patrice Philippon
Affiliation:
Probèmes Diophantiens UniversitéP. et M. Curie T.45–46, 5 ème ét. 75252 Paris, Cedex 05 FranceFrance
Rights & Permissions [Opens in a new window]

Extract

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.

We consider the classifying set, denoted ℂ/˜ below, introduced by Mahler and we show that it can be endowed with non-discrete, Hausdorff topologies (even local distances) based on diophantine approximation properties of (complex) numbers. We then establish several results on the pointed topological space obtained, which could be dubbed Mahler's space (of first degree). We recover an analogue of Mahler's original classification by considering local distances to the special point (that is the class of all algebraic numbers). The main ingredients used here are diophantine properties in dimension zero over ℚ and non-standard analysis.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 49 , Issue 2 , April 1994 , pp. 219 - 238
Copyright
Copyright © Australian Mathematical Society 1994

References

Références

[1]Amoroso, F., ‘On the distribution of complex numbers according to their transcendance types’, Ann. Mat. Pura Appl. (4) 151 (1988), 359–368.CrossRefGoogle Scholar
[2]Baker, A., Transcendental number theory (Cambridge Univ. Press, 1979).Google Scholar
[3]Bourbaki, N., Topologie générale, fasc. de résultats (Hermann, 1964).Google Scholar
[4]Durand, A., ‘Quatre problèmes de Mahler sur la fonction ordre d'un nombre transcendant’, Bull. Soc. Math. France 102 (1974), 365–377.CrossRefGoogle Scholar
[5]Durand, A., ‘Fonction θ–ordre et classification de ℂp’, C.R. Acad. Sci. Paris Sér. A Math. 280 (1975), 1085–1088.Google Scholar
[6]Hellegouarch, Y., ‘Introduction à la théorie des nombres non-standard’, in Séminaire de Théorie des Nombres, 1991 (Publ. Univ. Caen, 1991), pp. 10.Google Scholar
[7]Lindstrom, T., ‘An invitation to non-standard analysis’, in Nonstandard analysis and its applications, (Cutland, N.. Editor), London Math. Soc. Stud. Texts 10 (Cambridge Univ. Press, 1988), pp. 1–105.Google Scholar
[8]Mahler, K., ‘On the order function of a transcendental number’, Acta Arith. 18 (1971), 63–76.CrossRefGoogle Scholar
[9]Philippon, P., ‘Critères pour I'indépendance algébrique’, Inst. Hautes Etudes Sci. Publ. Math. 64 (1986), 5–52.CrossRefGoogle Scholar
[10]Schneider, T., Einführung in die Transzendenten Zahlen, Grundlehren Math. Wiss. 81 (Springer, Berlin, Heidelberg, New York, 1957). Introduction aux nombres transcendants (Gauthier-Villars, Paris, 1959).CrossRefGoogle Scholar
[11]Sprindzhuk, V.G., ‘Achievements and problems in diophantine approximation theory’, Uspekhi Mat. Nauk 35 (1980), 3–68. Russian Math. Surveys 35(4) (1980), 1–80.Google Scholar
[12]Waldschmidt, M., Nombres transcendants, Lecture Notes in Math. 402 (Springer, Berlin, Heidelberg, New York, 1974).CrossRefGoogle Scholar
[13]KunRui, Yu, ‘A generalization of Mahler's classification to several variables’, J. Reine Angew. Math. 377 (1987), 113–126.Google Scholar