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Bulletin of the London Mathematical Society Advance Access originally published online on April 25, 2008
Bulletin of the London Mathematical Society 2008 40(3):447-456; doi:10.1112/blms/bdn025
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© 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.


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