Pairs of homogeneous additive equations

Abstract

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.