Let [ ] be a [pfr ]-adic field, and consider the system F = (F1,…,FR) of diagonal equations

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[ges ]Γ*(k, R, [ ]). For example, when k = 1, ordinary linear algebra tells us that Γ*(1, R, [ ]) = R + 1 for any field [ ]. We also define Γ*(k, R) to be the smallest integer N such that Γ*(k, R, ℚp) [les ] N for all primes p.