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

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.

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Hardy–Littlewood method, diagonal forms, forms in many variables
Mathematical Subject Classification
Primary: 11D72
Secondary: 11D45, 11P55, 14G05
Received: 18 May 2020
Revised: 19 December 2020
Accepted: 23 February 2021
Published: 23 December 2021
Julia Brandes
University of Gothenburg and Chalmers University of Technology
Scott T. Parsell
West Chester University
West Chester, PA
United States