Optimal mean value estimates beyond Vinogradov's mean value theorem

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.Comment: Our original treatment of systems with degrees $k \ge 4$ contained a fatal flaw (thanks to S. T. Parsell for alerting us to this). The revised version gives an adapted treatment, leading to different results for $k \ge 4$. All results involving only quadratic and cubic equations remain unaffecte