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Large transcendence degree

Published online by Cambridge University Press:  09 April 2009

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

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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.

Information

Type
Research Article
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. 149173 (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. 175188 (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), 239247.Google Scholar
[6]Diaz, G., ‘Grand degrés de transcendance pour des families d'exponentielles’, J. Number Theory 31 (1) (01 1989), 123.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), 295302.Google Scholar
[8]Masser, D. W., ‘On polynomials and exponential polynomials in several complex variables’, Invent. Math. 63 (1981), 8195.Google Scholar
[9]Masser, D. W., Wüstholz, G., ‘Fields of large transcendence degree generated by values of elliptic functions’, Invent. Math. 72 (1983), 407464.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), 355383.Google Scholar
[12]Reyssat, E., ‘Approximation algébrique de nombres liés aux fonctions elliptiques et exponentielles’, Bull. Soc. Math. France 108 (1980), 4779.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), 279307.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. 942967 (Walter de Gruyter).Google Scholar
[16]Tubbs, R., ‘A diophantine problem on elliptic curves’, Trans. Amer. Math. Soc. 309 (1) (1988), 325338.CrossRefGoogle Scholar
[17]Waldschmidt, M., ‘Groupes algébreiques et grands degrés de transcendance’, Acta Math. 156 (1986), 253302.CrossRefGoogle Scholar