Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T01:31:12.225Z Has data issue: false hasContentIssue false
  • English
  • Français

Large transcendence degree

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  09 April 2009

Robert Tubbs
Robert Tubbs
Affiliation:
University of ColoradoBoulder, Colorado 80309-0426, U.S.A.
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.

In this paper we study the transcendence degree of fields generated over Q by the numbers associated with values of one-parameter subgroups of commutative algebraic groups. We show that in many instances these fields have a large transcendence degree when measured in terms of the available data.

Our method deals with points which are “well distributed” (in a sense which is made precise) among certain algebraic subgroups of the algebraic group under consideration. We verify that these results apply in many classical situations.

MSC classification

Secondary: 11J99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 51 , Issue 3 , December 1991 , pp. 400 - 422
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Ably, M., Independance algebrique pour les groupes algebriques commutafs, (Thèse, Lille, 1989), p. 75.Google Scholar
[2]Bertrand, D., ‘Minimal heights and polarizations on abelian varieties’, Math. Sciences Research Institute, Berkeley, California (06 1987), p. 18.Google Scholar
[3]Brownawell, W. D., ‘Large transcendence degree revisited I. Exponential and non-CM cases’, Diophantine Approximation and Transcendence Theory, Seminar Bonn edited by Wüstholz, G., pp. 149–173 (Springer Lecture Notes 1290, 1985).Google Scholar
[4]Brownawell, W. D., Tubbs, R., ‘Large transcendence degree revisited II. The CM case’, Diophantine Approximation and Transcendence Theory, Seminar Bonn edited by Wüstholz, G., pp. 175–188 (Springer Lecture Notes 1290, 1985).Google Scholar
[5]Cisjouw, P. L. and Tijdemann, R., ‘An auxiliary result in the theory of transcendence numbers’, Duke Math. J. 42 (2) (1975), 239–247.Google Scholar
[6]Diaz, G., ‘Grand degrés de transcendance pour des families d'exponentielles’, J. Number Theory 31 (1) (01 1989), 1–23.CrossRefGoogle Scholar
[7]Fresnel, J., ‘Déformation d'un groupe algébrique’, Appendice in ‘Groupes algébriques et grands degrés de transcendance’, Waldschmidt, M., Acta Math. 156 (1986), 295–302.Google Scholar
[8]Masser, D. W., ‘On polynomials and exponential polynomials in several complex variables’, Invent. Math. 63 (1981), 81–95.Google Scholar
[9]Masser, D. W., Wüstholz, G., ‘Fields of large transcendence degree generated by values of elliptic functions’, Invent. Math. 72 (1983), 407–464.CrossRefGoogle Scholar
[10]Philippon, P., ‘Criteres pour l'independance algébrique’, Inst. Hautes Études Sci. Publ. Math. 64.Google Scholar
[11]Philippon, P., ‘Un lemme de zéros dans les groupes algébriques commutatifs’, Bull. Soc. Math. France 114 (1986), 355–383.Google Scholar
[12]Reyssat, E., ‘Approximation algébrique de nombres liés aux fonctions elliptiques et exponentielles’, Bull. Soc. Math. France 108 (1980), 47–79.CrossRefGoogle Scholar
[13]Serre, J. P., ‘Quelques propriétés des groupes algébriques commutatifs’. Appendice II, Nombres Transcendants et Groupes Algébriques, edited by Waldschmidt, M. (Astérisque, No. 69–70).Google Scholar
[14]Tubbs, R., ‘Algebraic groups and small transcendence degree I’, J. Number Theory 25 (3) (1987), 279–307.CrossRefGoogle Scholar
[15]Tubbs, R., ‘The algebraic independence of three numbers: Schneider's method’, Proceedings of the Quebec Conference, edited by DeKoninck, J. M. and Levesque, C., pp. 942–967 (Walter de Gruyter).Google Scholar
[16]Tubbs, R., ‘A diophantine problem on elliptic curves’, Trans. Amer. Math. Soc. 309 (1) (1988), 325–338.CrossRefGoogle Scholar
[17]Waldschmidt, M., ‘Groupes algébreiques et grands degrés de transcendance’, Acta Math. 156 (1986), 253–302.CrossRefGoogle Scholar