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 > (3 r + 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 > r 2 k .