Cubic moments of Fourier coefficients and pairs of diagonal quartic forms

TL;DR

This paper proves the non-singular Hasse Principle for pairs of diagonal quartic equations in 22 or more variables using advanced moment estimation techniques involving Fourier coefficients.

Contribution

It introduces a novel cubic moment approach to estimate Fourier coefficients, enabling the proof of the Hasse Principle for these equations.

Findings

Established the non-singular Hasse Principle for pairs of diagonal quartic equations in 22+ variables

Developed a new method involving cubic moments of Fourier coefficients

Achieved bounds on entangled moments of quartic Weyl sums

Abstract

We establish the non-singular Hasse Principle for pairs of diagonal quartic equations in 22 or more variables. Our methods involve the estimation of a certain entangled two-dimensional 21st moment of quartic smooth Weyl sums via a novel cubic moment of Fourier coefficients.