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

Abstract

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.