Counting rational points on biquadratic hypersurfaces

TL;DR

This paper develops an asymptotic formula for counting rational points of bounded height on smooth biquadratic hypersurfaces using the Hardy--Littlewood circle method, advancing understanding in Diophantine geometry.

Contribution

It provides the first asymptotic count for rational points on arbitrary smooth biquadratic hypersurfaces in many variables.

Findings

Established an asymptotic formula for rational points

Applied Hardy--Littlewood circle method successfully

Extended results to a broad class of hypersurfaces

Abstract

An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariski open subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables. The proof uses the Hardy--Littlewood circle method.