Rational points on varieties defined by multihomogeneous diagonal forms

TL;DR

This paper derives an asymptotic count of rational points on certain algebraic varieties defined by multihomogeneous diagonal equations, employing advanced analytic number theory techniques.

Contribution

It introduces a new asymptotic formula for rational points on these varieties using the Hardy-Littlewood circle method and hyperbola method.

Findings

Asymptotic formula for rational points established

Effective bounds on the number of solutions obtained

Method extends previous approaches to multihomogeneous systems

Abstract

We give an asymptotic formula for the number of rational points of bounded height on algebraic varieties defined by systems of multihomogeneous diagonal equations. The proof uses the Hardy-Littlewood circle method and the hyperbola method developed by Blomer and Br\"udern.