It is shown that the system of two additive equations a1x1k+…+asxsk=b1x1k+…+bsxsk=0 where k ⩾ 2 and aj, bj are any given integers, has non‐trivial solutions in all p‐adic fields provided only that s > 8k2. The constant 8 can be reduced when k is not a power of 2. It is expected, in accordance with a classical conjecture of Artin, that the bound 8k2 can be replaced by 2k2.
2000 Mathematical Subject Classification: 11D72.
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]
with integer coefficients aij, has a non‐trivial solution in Qp for all primes p whenever
[formula]
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]
are sufficient. There have been a number of refinements of these results. Schmidt [13] obtained N≫R2k3 log k, and Low, Pitman and Wolff [10] improved the work of Davenport and Lewis by showing the weaker constraints
[formula]
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. 1991 Mathematics Subject Classification 11D72, 11D79.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.