Optimal mean value estimates beyond Vinogradov's mean value theorem

2019/01/10 by Brandes, Julia, Wooley, Trevor D.
#11D45 #11L07 #11L15 #11P55 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1901.03153

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 ≥ 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.

Related

• Mean value estimates for Weyl sums in two dimensions
• Prime Solutions of Diagonal Diophantine Systems
• The Cubic Case of Vinogradov's Mean Value Theorem --- A Simplified Approach to Wooley's "Efficient Congruencing"
• Integral solutions to systems of diagonal equations
• The Hasse principle for systems of quadratic and cubic diagonal equations