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
Milestones
Received: 18 May 2020
Revised: 19 December 2020
Accepted: 23 February 2021
Published: 23 December 2021
Authors
Julia Brandes
University of Gothenburg and Chalmers University of Technology
Gothenburg
Sweden
Scott T. Parsell
West Chester University
West Chester, PA
United States