The density of rational points on Cayley's cubic surface

TL;DR

This paper confirms that the number of primitive integer solutions on Cayley's cubic surface grows proportionally to B(log B)^6, aligning with Manin's conjecture, thus advancing understanding of rational points on algebraic surfaces.

Contribution

It provides a precise asymptotic count for primitive integer points on Cayley's cubic surface, verifying a prediction of Manin's conjecture.

Findings

Number of primitive integer points is of order B(log B)^6

Results match the prediction of Manin's conjecture

Advances understanding of rational points on algebraic surfaces

Abstract

The Cayley cubic surface is given by the equation sum_{i=1}^4 X_i^{-1}=0. We show that the number of non-trivial primitive integer points of size at most B is of exact order B(log B)^6, as predicted by Manin's conjecture.