Simultaneous additive equations: Repeated and differing degrees

TL;DR

This paper establishes bounds on the number of variables needed for Hasse principles in systems of additive equations with mixed degrees, advancing understanding of hybrid systems beyond previous weak estimates.

Contribution

It provides the first significant bounds for hybrid systems of additive equations with forms of differing degrees, extending beyond prior general estimates.

Findings

Bounds for variable requirements in hybrid systems

First results on systems with mixed degrees

Improved understanding of Hasse principles for such systems

Abstract

We obtain bounds for the number of variables required to establish Hasse principles, both for existence of solutions and for asymptotic formulae, for systems of additive equations containing forms of differing degree but also multiple forms of like degree. Apart from the very general estimates of Schmidt and Browning--Heath-Brown, which give weak results when specialized to the diagonal situation, this is the first result on such "hybrid" systems.