Hua's lemma and simultaneous diagonal equations

This paper concerns systems of r homogeneous diagonal equations of degree k in s variables, with integer coefficients. Subject to a suitable non-singularity condition, it is shown that the expected asymptotic formula holds for the number of such systems inside a box [-P, P](s), provided only that s > (3r + 1)2(k-2). By way of comparison, classical methods based on the use of Hua's lemma would establish a similar conclusion, provided instead that s > r2(k).