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A generalization of Vinogradov's mean value theorem

Published online by Cambridge University Press:  22 June 2005

Scott T. Parsell
Scott T. Parsell
Affiliation:
Department of Mathematics and Actuarial Science, Butler University, 4600 Sunset Avenue, JH 270, Indianapolis, IN 46208, USA. E-mail: sparsell@butler.edu
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Abstract

We obtain new upper bounds for the number of integral solutions of a complete system of symmetric equations, which may be viewed as a multi-dimensional version of the system considered in Vinogradov's mean value theorem. We then use these bounds to obtain Weyl-type estimates for an associated exponential sum in several variables. Finally, we apply the Hardy–Littlewood method to obtain asymptotic formulas for the number of solutions of the Vinogradov-type system and also of a related system connected to the problem of finding linear spaces on hypersurfaces.

Keywords

counting solutions of diophantine equationsexponential sumsapplications of the Hardy–Littlewood method11D4511D7211L0711P55
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 91 , Issue 1 , July 2005 , pp. 1 - 32
Copyright
2005 London Mathematical Society

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Footnotes

Research supported in part by a National Science Foundation Postdoctoral Fellowship (DMS-0102068).