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On Pairs of Diagonal Quintic Forms

Published online by Cambridge University Press:  04 December 2007

Scott T. Parsell  and
Trevor D. Wooley
Scott T. Parsell
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, U.S.A. E-mail: [email protected]
Trevor D. Wooley
Affiliation:
Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, MI 48109-1109, U.S.A. E-mail: [email protected]
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Abstract

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We demonstrate that a pair of additive quintic equations in at least 34 variables has a nontrivial integral solution, subject only to an 11-adic solubility hypothesis. This is achieved by an application of the Hardy–Littlewood method, for which we require a sharp estimate for a 33.998th moment of quintic exponential sums. We are able to employ p-adic iteration in a form that allows the estimation of such a mean value over a complete unit square, thereby providing an approach that is technically simpler than those of previous workers and flexible enough to be applied to related problems.

Keywords

diophantine equationsquintic formsthe Hardy–Littlewood method
Type
Research Article
Information
Compositio Mathematica , Volume 131 , Issue 1 , March 2002 , pp. 61 - 96
Copyright
© 2002 Kluwer Academic Publishers