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

TL;DR

This paper investigates a specific class of Diophantine systems related to Vinogradov's mean value theorem, demonstrating a scarcity of non-diagonal positive integer solutions, especially when certain parameters are small relative to k.

Contribution

It introduces new bounds on the number of solutions for a family of Diophantine systems related to Vinogradov's theorem, extending understanding of solution scarcity.

Findings

Non-diagonal solutions are sparse for the considered systems.

Quantitative bounds are especially sharp when d=o(k^{1/4}).

Results generalize aspects 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 $d=o(k^{1/4})$.