Vinogradov systems with a slice off

Journal article, 2017

Let I-s,I-k,I-r(X) denote the number of integral solutions of the modified Vinogradov system of equations with 1 <= x(i),y(i) <= X (1 <= i <= s). By exploiting sharp estimates for an auxiliary mean value, we obtain bounds for I-s,I-k,I-r(X) for 1 <= r <= k - 1. In particular, when s,k is an element of N satisfy k >= 3 and 1 <= s <= (k(2) - 1)/2, we establish the essentially diagonal behaviour I-s,I-k,I-l(X) << Xs+epsilon.