Vinogradov systems with a slice off

TL;DR

This paper investigates a modified Vinogradov system with a missing degree, deriving bounds for the number of solutions and demonstrating near-diagonal behavior under certain conditions.

Contribution

It introduces bounds for solutions to a modified Vinogradov system with a missing degree, extending previous estimates and establishing near-diagonal solution counts.

Findings

Established bounds for $I_{s,k,r}(X)$ for $1 extless r extless k$.

Proved essentially diagonal behavior $I_{s,k,1}(X) \\ll X^{s+\\epsilon}$ for certain $s,k$.

Extended mean value estimates to analyze solutions of the modified system.

Abstract

Let $I_{s,k,r}(X)$ denote the number of integral solutions of the modified Vinogradov system of equations $$x_1^j+\ldots +x_s^j=y_1^j+\ldots +y_s^j\quad (\text{$1\le j\le k$, $j\ne r$}),$$ with $1\le x_i,y_i\le X$ $(1\le i\le s)$. By exploiting sharp estimates for an auxiliary mean value, we obtain bounds for $I_{s,k,r}(X)$ for $1\le r\le k-1$. In particular, when $s,k\in \mathbb N$ satisfy $k\ge 3$ and $1\le s\le (k^2-1)/2$, we establish the essentially diagonal behaviour $I_{s,k,1}(X)\ll X^{s+\epsilon}$.