On Artin's Conjecture, I: Systems of Diagonal Forms

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 a ij , has a non‐trivial solution in Q p for all primes p whenever [formula] Here we adopt the convention that a solution of (1) is non‐trivial if not all the x i 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 ≫ R 2 k 3 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 k 3 log k , in spite of the expected k 2 in (2). In this paper we show that one can reach the expected order of magnitude k 2 . 1991 Mathematics Subject Classification 11D72, 11D79.