Skip Navigation

Proceedings of the London Mathematical Society 2005 91(1):1-32; doi:10.1112/S002461150501525X
This Article
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Parsell, S. T.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© London Mathematical Society

A Generalization of Vinogradov's Mean Value Theorem

Scott T. Parsell

Department of Mathematics and Actuarial Science, Butler University 4600 Sunset Avenue, JH 270, Indianapolis, IN 46208, USA. E-mail: sparsell{at}butler.edu

Received 5 February 2004. Revision received 17 November 2004.

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. 2000 Mathematics Subject Classification 11D45, 11D72, 11L07, 11P55.

Key Words: counting solutions of diophantine equations • exponential sums • applications of the Hardy–Littlewood method

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


Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?




Disclaimer: Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department.