Paucity problems and some relatives of Vinogradov’s mean value theorem

Abstract

Abstract When $k\geqslant 4$ and $0\leqslant d\leqslant (k-2)/4$ , we consider the system of Diophantine equations \begin{align*}x_1^j+\ldots +x_k^j=y_1^j+\ldots +y_k^j\quad (1\leqslant j\leqslant k,\, j\ne k-d).\end{align*} We show that in this cousin of a Vinogradov system, there is a paucity of non-diagonal positive integral solutions. Our quantitative estimates are particularly sharp when $d=o\!\left(k^{1/4}\right)$ .