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

Brandes, Julia; Parsell, Scott T.

doi:10.2140/ant.2021.15.2289

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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

DOI: 10.2140/ant.2021.15.2289

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

Vol. 15, No. 9, 2021