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On pairs of cubic Diophantine inequalities

Part of: Diophantine equations

Published online by Cambridge University Press:  26 February 2010

J. Brüdern  and
R. J. Cook
J. Brüdern
Affiliation:
Mathematisches Institut, Bunsenstrasse 3-5, 3400 Göttingen, Germany.
R. J. Cook
Affiliation:
Dept. of Pure Mathematics, University of Sheffield Hicks Building, Sheffield S3 7RH, U.K..
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Extract

The problem of finding rational points on varieties defined by two additive cubic equations has attracted some interest. Davenport and Lewis [12], Cook [8] and Vaughan [16] showed that the pair of equations

with integer coefficients a,, bt always has a nontrivial solution when s = 18, s = 17, and 5 = 16 respectively. Vaughan's result in s = 16 variables is best possible since there are examples of pairs of equations (1) with s = 15 which fail to vanish simultaneously in the 7-adic field. However if the existence of a 7-adic solution is assured then Baker and Briidern [2], building on work of Cook [9], showed that s = 16 could be replaced by s = 15, and recently Briidern [5] has obtained the result with s = 14.

MSC classification

Secondary: 11D75: Diophantine inequalities
Type
Research Article
Information
Mathematika , Volume 38 , Issue 2 , December 1991 , pp. 250 - 263
Copyright
Copyright © University College London 1991

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