Systems of Diagonal Equations over J-adic Fields

with coefficients in +. It is an interesting problem in number theory to determine when such a system possesses a nontrivial +-rational solution. In particular, we define Γ*(k,R,+) to be the smallest natural number such that any system of R equations of degree k in N variables with coefficients in + has a nontrivial +-rational solution provided only that N&Γ*(k,R,+). For example, when k ̄ 1, ordinary linear algebra tells us that Γ*(1,R,+) ̄R­1 for any field +. We also define Γ*(k,R) to be the smallest integer N such that Γ*(k,R,1 p )%N for all primes p. When + ̄1 p , much is known about this problem. In the case where R ̄ 1, Davenport and Lewis [5] showed that Γ*(k, 1)%k#­1 for each k, with equality holding whenever k ̄ p®1 for some prime p. When R ̄ 2 and k is odd, Davenport and Lewis [6] showed that Γ*(k, 2)% 2k#­1. For general R, a conjecture of Artin’s suggests that one should have Γ*(k,R)%Rk#­1, but this is not known in any case other than the three above. Despite the inability to obtain the conjectured bound, several authors have found upper bounds for Γ*(k,R). Davenport and Lewis [7] obtained the bound Γ*(k,R)% [9R#k log(3Rk)] for all odd k, and the bound Γ*(k,R)% [48R#k$ log(3Rk#)] for all even k larger than 2. This was improved in most cases by Low, Pitman and Wolff [11], who showed that the bound Γ*(k,R)% [48Rk$ log(3Rk#)] is sufficient for all k larger than 2, and that the bound Γ*(k,R)% 2R#k logk holds whenever k is odd and sufficiently large. Recently, Bru$ dern and Godinho [3] obtained the bound Γ*(k,R)%R$k# whenever R and k are at least 3, except for the case in which R ̄ 3 and k is a power of 2, when one has Γ*(k, 3)% 36k#. This bound is better than those of Low, Pitman and Wolff and Davenport and Lewis when k is even and suitably large compared with R. Also, this result is notable because it shows that a bound of the form Γ*(k,R)' R k# is possible for all values of k. The primary purpose of this paper is to make an improvement on the bound of Bru$ dern and Godinho through methods involving the use of Teichmu$ ller representatives. To this end, we will prove the following theorem.