In this paper it is proved that the pair f =a 1 x k 1 + } } } +a n x k n , g=b 1 x k 1 + } } } + b n x k n , with k= p { ( p&1) k 0 and (k 0 , p)=1, has a common p-adic zero provided n 2k 2+w( p, {) , where w( p, {)=1ร(log p+(1ร{) log( p&1)). It is also proved that if n 2k 2 , then the pair f, g has a common p-adic zero provided that k 0 2 { .
1998 Academic Press
The question of finding p-adic zeros (non-trivial solutions) for a pair of additive forms
with rational coefficients, was first addressed by Davenport and Lewis in [6], then generally discussed in . They have proved that Theorem 0.1. If n 2k 2 +1 and k is odd then the pair in (1) has p-adic zeros for all p. This is a confirmation of a longstanding conjecture of E. Artin for the case of additive forms of odd degree. For the case of k even they proved that p-adic solubility occurs for all primes p if n 7k 3 . They also remarked (in fact proved) that for k even and n 2k 2 +1, p-adic solubility occurs for all primes but the prime divisors p of k, such that p( p&1) also divides k, especially for p=2 and k=2 l (we will refer to these primes as singular primes). In 1989, Atkinson and Cook proved (see ) that if p>k 6 then one only needs n 4k+1 to have p-adic solubility (this paper has a nice introduction to the subject and many references), which stress the fact that the farther we get from the prime divisors of k, especially the singular primes, the fewer variables we need to guarantee p-adic solubility.