As a special case of a well-known conjecture of Artin, it is expected that a system
of R additive forms of degree k, say
formula here
with integer coefficients aij, has a
non-trivial solution in ℚp for all primes p whenever
formula here
Here we adopt the convention that a solution of (1) is non-trivial if not all the
xi are 0. To date, this has been verified only
when R=1, by Davenport and Lewis
[4], and for odd k when R=2,
by Davenport and Lewis [7]. For larger values of
R, and in particular when k is even, more severe conditions
on N are required to assure the existence of p-adic solutions of
(1) for all primes p. In another important
contribution, Davenport and Lewis [6] showed that the conditions
formula here
are sufficient. There have been a number of refinements of these results. Schmidt
[13] obtained
N[Gt ]R2k3 log k,
and Low, Pitman and Wolff [10] improved the work
of Davenport and Lewis by showing the weaker constraints
formula here
to be sufficient for p-adic solubility of (1).
A noticeable feature of these results is that for even k, one always encounters a
factor k3 log k, in spite of the expected k2 in (2). In this paper we show that one can
reach the expected order of magnitude k2.