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

Abstract

When $k\ge 4$ and $0\le d\le (k-2)/4$, we consider the system of Diophantine equations \[ x_1^j+\ldots +x_k^j=y_1^j+\ldots +y_k^j\quad (1\le j\le k,\, j\ne k-d). \] 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 $d=o(k^{1/4})$.

17 pages

References in corpus (2)

• The cubic case of the main conjecture in Vinogradov's mean value theorem
• Nested efficient congruencing and relatives of Vinogradov's mean value theorem

Cited by in corpus (1)

• The paucity problem for certain symmetric Diophantine equations