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Additive relations in fields

Part of: Diophantine equations Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  09 April 2009

A. J. Van Der Poorten  and
H. P. Schlickewei
A. J. Van Der Poorten
Affiliation:
School of Mathematics, Physics, Computing and Electronics Macquarie University, NSW 2109 Australia
H. P. Schlickewei
Affiliation:
Abteilung für Mathematik Universität UlmOberer Eselsberg, Postfach 4066 D-7900 Ulm Germany
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Abstract

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This paper is the first part of a long delayed revision of the manuscript ‘The growth conditions recurrence sequences’ (circulated in 1982) in which the authors outlined a proof of the now well known theorem on the finiteness of the number of solutions of S-unit equations. The argument lifting the result from number fields to arbitrary fields of characteristic zero has original features.

MSC classification

Secondary: 11J25: Diophantine inequalities 11D72: Equations in many variables

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 51 , Issue 1 , August 1991 , pp. 154 - 170
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Brownawell, W. D. and Masser, D. W., ‘Vanishing sums in function fields’, Math. Proc. Cambridge Phil. Soc. 100 (1986), 427–434.CrossRefGoogle Scholar
[2]Cassels, J. W. S., ‘An embedding theorem for fields’, Bull. Austral. Math. Soc. 14 (1976) 193–198; Addendum, Bull. Austral. Math. Soc. 14 (1976) 479–480.Google Scholar
[3]Dubois, E. and Rhin, G., ‘Sur la majoration de formes linéaires à coefficients algébriques réels et p-adiques (Démonstration d'une conjecture de K. Mahler)’, C. R. Acad. Sci. Paris A282 (1976), 1211.Google Scholar
[4]Jan-Hendrik, Evertse, ‘On sums of S-units and linear recurrences’, Compositio Math. 53 (1984), 225–244.Google Scholar
[5]Evertse, J.-H. and Győry, K, ‘On the number of solutions of weighted unit equations’, Compositio Math. 66 (1988), 329–354.Google Scholar
[6]Loxton, J. H. and van der Poorten, A. J., ‘Multiplicative dependence in number fields’, Acta Arith. 42 (1983), 291–302.CrossRefGoogle Scholar
[7]Mahler, K., ‘Zur Approximation algebraiseher Zahien I (Über den groβten Primteiler binärer Formen)’, Math. Ann. 107 (1933), 691–730.CrossRefGoogle Scholar
[8]Mason, R. C., Diophantine equations over function fields, London Math. Soc. Lecture Notes 96, Cambridge University Press (1984).CrossRefGoogle Scholar
[9]Masser, D. W., ‘Linear relations on algebraic groups’ in Baker, Alan, (Ed.), New Advances in Transcendence Theory, Cambridge University Press (1988), 248–262.Google Scholar
[10]van der Poorten, A. J., ‘Some problems of recurrent interest’, Colloq. Math. Soc. János Bolyai (Budapest, 1981) 34, Topics in Number Theory, North-Holland (1984), 1265–1294.Google Scholar
[11]van der Poorten, A. J. and Schlickewei, H. P., ‘The growth conditions for recurrence sequences’, Macquarie Math. Reports 82–0041 (08, 1982), Macquarie University, Australia2109.Google Scholar
[12]van der Poorten, A. J., ‘Some facts that should be better known; especially about rational functions’, in Mollin, R. A. (Ed.), Number Theory and Applications, (NATO-Advanced Study Institute, Banff 1988), Kluwer Academic Publishers, Dordrecht (1989), 497–528.Google Scholar
[13]Schlickewei, H. P., ‘Linearformen mit algebraischen Koeffizienten’, Manuscripta Math. 18 (1976), 147–185.Google Scholar
[14]Schlickewei, H. P., ‘Über die diophantisehe Gleichung x 1 + x 2 + … + xn = 0 ’, Acta Arith. 33 (1977), 183–185.CrossRefGoogle Scholar
[15]Schlickewei, H. P., ‘The p-adic Thue-Siegel-Roth-Schmidt theorem’, Arch. Mat. 29 (1977), 267–270.CrossRefGoogle Scholar
[16]Schmidt, W. M., ‘Simultaneous approximation to algebraic numbers by elements of a number field’, Monatsh. Math. 79 (1975), 55–66.Google Scholar
[17]Schmidt, W. M., Diophantine approximation, Lecture Notes in Math. 785, Springer-Verlag (1980).Google Scholar
[18]Vojta, P. A., Integral points on varieties, PhD Thesis, Harvard University (1983).Google Scholar