The Hasse principle for systems of quadratic and cubic diagonal equations

TL;DR

This paper proves an asymptotic Hasse principle for systems of quadratic and cubic diagonal equations, using new methods to reduce the number of variables needed and achieve near square root cancellation.

Contribution

It introduces a novel application of Br"udern's and Wooley's complification method to systems of quadratic and cubic forms, lowering variable requirements.

Findings

Established asymptotic Hasse principle for specified systems

Reduced the number of variables needed compared to previous work

Achieved near square root cancellation for certain systems

Abstract

Employing Br\"udern's and Wooley's new complification method, we establish an asymptotic Hasse principle for the number of solutions to a system of r_3 cubic and r_2 quadratic diagonal forms, when the number of cubic equations is at least double the number of quadratic forms. The number of variables required is lower than in earlier work. In particular, we achieve essentially square root cancellation for systems of one quadratic and arbitrarily many cubic equations.