Skip Navigation



Bulletin of the London Mathematical Society Advance Access published online on April 25, 2008
Bulletin of the London Mathematical Society, doi:10.1112/blms/bdn025
This Article
Right arrow FREE Full Text (PDF) Freely available
Right arrow All Versions of this Article:
40/3/447    most recent
bdn025v1
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 Knapp, M. P.
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© 2008 London Mathematical Society

Pairs of homogeneous additive equations

Michael P. Knapp

Mathematical Sciences Department
Loyola College
4501 North Charles Street
Baltimore, MD 21210-2699
USA

Received 2 June 2007. Revision received 8 January 2008.

In 1966, Davenport and Lewis published their paper ‘Notes on congruences III’, in which they proved that under some mild conditions a system of two additive forms of equal degrees must have a nonsingular simultaneous zero modulo any prime number. In their paper, they asked whether the theorem is true in general finite fields and pointed out that one of their key lemmas is no longer true in this situation. In this paper we answer their question in the affirmative, proving that under the same conditions a system of two additive forms over any finite field must have a nonsingular simultaneous zero. We then apply this result to obtain an upper bound on the number of variables required to ensure that a system of two additive forms of equal degree has a nontrivial zero in a p-adic field.

2000 Mathematics Subject Classification 11D72 (primary), 11T06, 11E95 (secondary).

The work is supported by NSF grant DMS-0344082.


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.