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 here

with integer coefficients a_ij, has a non-trivial solution in ℚ_p for all primes p whenever

formula here

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 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.