Integral points of bounded height on a certain toric variety

TL;DR

This paper derives an asymptotic count of integral points of bounded height on a specific toric variety, introduces a new interpretation of the formula, and relates it to a novel obstruction affecting the density of integral points.

Contribution

It provides a new asymptotic formula for integral points on a toric variety, along with an analogue of Peyre's constant and a new obstruction to Zariski density.

Findings

Asymptotic formula for integral points established

An analogue of Peyre's constant constructed

Identification of a new obstruction to Zariski density

Abstract

We determine an asymptotic formula for the number of integral points of bounded height on a certain toric variety, which is incompatible with part of a preprint by Chambert-Loir and Tschinkel. We provide an alternative interpretation of the asymptotic formula we get. To do so, we construct an analogue of Peyre's constant $\alpha$ and describe its relation to a new obstruction to the Zariski density of integral points in certain regions of varieties.