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

Abstract

AbstractWhen$k\geqslant 4$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$d=o\!\left(k^{1/4}\right)$.