Rational points on the inner product cone via the hyperbola method

TL;DR

This paper proves a strong quantitative version of Manin's conjecture for a specific biprojective variety, utilizing the circle method and evaluating a complex singular integral explicitly.

Contribution

It introduces a novel approach to Manin's conjecture for biprojective varieties and computes a complex singular integral in closed form.

Findings

Established a strong quantitative form of Manin's conjecture for a specific variety.

Evaluated the singular integral involving the third power of the sine function explicitly.

Demonstrated the effectiveness of the circle method in this context.

Abstract

A strong quantitative form of Manin's conjecture is established for a certain variety in biprojective space. The singular integral in an application of the circle method involves the third power of the integral sine function, and is evaluated in closed form.