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

As a special case of a well-known conjecture of Artin, it isexpected that a system of R additive forms of degree k, say formula] with integer coefficients a_ij, has a non-trivial solution inQ_p for all primes p whenever formula] Here we adopt the convention that a solution of (1) is non-trivialif not all the x_i are 0. To date, this has been verified onlywhen 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 particularwhen k is even, more severe conditions on N are required toassure the existence of p-adic solutions of (1) for all primesp. In another important contribution, Davenport and Lewis 6]showed that the conditions formula] are sufficient. There have been a number of refinements of theseresults. Schmidt 13] obtained N>>R²k³ log k, and Low,Pitman and Wolff 10] improved the work of Davenport and Lewisby 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, onealways encounters a factor k³ log k, in spite of the expectedk² in (2). In this paper we show that one can reach the expectedorder of magnitude k². 1991 Mathematics Subject Classification11D72, 11D79.