Vinogradov systems with a slice off

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 $I_{s,k,1}(X)\ll X^{s+\epsilon}$.