The Hasse principle for systems of diagonal cubic forms

TL;DR

This paper proves the Hasse principle for systems of diagonal cubic equations under specific conditions, extending the applicability of the circle method to the optimal variable range.

Contribution

It establishes the Hasse principle for diagonal cubic systems with more than 6r variables and non-singular coefficient matrices, reaching the theoretical limit of the circle method.

Findings

Hasse principle holds for systems with >6r variables

Applicable when coefficient matrix has no singular r x r submatrix

Achieves the theoretical limit of the circle method for these systems

Abstract

We establish the Hasse Principle for systems of r simultaneous diagonal cubic equations whenever the number of variables exceeds 6r and the associated coefficient matrix contains no singular r x r submatrix, thereby achieving the theoretical limit of the circle method for such systems.