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On Pairs of Diagonal Quintic Forms

Published online by Cambridge University Press:  04 December 2007

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.

Type
Research Article
Copyright
© 2002 Kluwer Academic Publishers