When \(s\geqslant k\geqslant 3\) and \(n_1,\ldots ,n_k\) are large natural numbers, denote by \(A_{s,k}({\textbf{n}})\) the number of solutions in non-negative integers \({\textbf{x}}\) to the system \(\begin{aligned} x_1^j+\cdots +x_s^j=n_j\quad (1\leqslant j\leqslant k). \end{aligned}\) Under appropriate local solubility conditions on \({\textbf{n}}\) , we obtain an asymptotic formula for \(A_{s,k}({\textbf{n}})\) when \(s\geqslant k(k+1)\) . This establishes a local–global principle in the Hilbert–Kamke problem at the convexity barrier. Our arguments involve minor arc estimates going beyond square-root cancellation.
