Vinogradov systems with a slice off
Julia Brandes, Trevor D. Wooley

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 denote the number of integral solutions of the modified Vinogradov system of equations with . By exploiting sharp estimates for an auxiliary mean value, we obtain bounds for for . In particular, when satisfy and , we establish the essentially diagonal behaviour .
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