The Hasse principle for diagonal forms restricted to lower-degree hypersurfaces

We establish the analytic Hasse principle for Diophantine systems consisting of one diagonal form degree $k$ and general $d$, where $d$ is smaller than $k$. By employing a hybrid method that combines ideas from study forms with techniques adapted to case, we are able obtain bounds grow exponentially in but only quadratically $k$, reflecting growth rates typically obtained both problems separately. also discuss some most interesting generalisations our approach.