#### Vol. 15, No. 9, 2021

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

### Julia Brandes and Scott T. Parsell

Vol. 15 (2021), No. 9, 2289–2314
##### Abstract

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

##### Keywords
Hardy–Littlewood method, diagonal forms, forms in many variables
##### Mathematical Subject Classification
Primary: 11D72
Secondary: 11D45, 11P55, 14G05