A remark on the mean value inequalities related to Waring's problem

TL;DR

This paper investigates mean value inequalities related to Waring's problem by establishing sharp estimates for exponential sums involving Weyl's sums, under certain pointwise assumptions.

Contribution

It provides new sharp mean value estimates for exponential sums connected to Waring's problem, assuming specific pointwise bounds on Weyl's sums.

Findings

Established sharp mean value inequalities for exponential sums.

Linked mean value estimates to pointwise bounds on Weyl's sums.

Contributed to understanding of Waring's problem through exponential sum analysis.

Abstract

Assuming some pointwise estimates on certain Weyl's sum, we prove the sharp estimates of the mean value associated to the following exponential sum $$ \sum_{n=1}^N e^{2\pi i tn^d +2\pi i xn}\,. $$