# Optimal mean value estimates beyond Vinogradov's mean value theorem

## TL;DR

This paper presents new, sharper mean value estimates for certain Diophantine systems, achieving the best possible bounds and establishing the Hasse principle for specific cubic and quadratic equations, surpassing previous limitations.

## Contribution

It introduces the first bounds of this quality for non-Vinogradov type Diophantine systems and proves the Hasse principle in new parameter ranges.

## Key findings

- Attained sharpest conjectured bounds for certain Diophantine systems.
- Established the Hasse principle for systems with cubic and quadratic equations in specific variables.
- Achieved the convexity barrier for these problems.

## Abstract

We establish improved mean value estimates associated with the number of integer solutions of certain systems of diagonal equations, in some instances attaining the sharpest conjectured conclusions. This is the first occasion on which bounds of this quality have been attained for Diophantine systems not of Vinogradov type. As a consequence of this progress, whenever $u \ge 3v$ we obtain the Hasse principle for systems consisting of $v$ cubic and $u$ quadratic diagonal equations in $6v+4u+1$ variables, thus attaining the convexity barrier for this problem.