On the Zariski-Density of Integral Points on a Complement of Hyperplanes in P^n

TL;DR

This paper characterizes when S-integral points are not Zariski-dense on the complement of hyperplanes in projective space, depending on the hyperplanes and the number field, extending classical results for archimedean places.

Contribution

It provides a complete characterization of Zariski-density of S-integral points in the complement of hyperplanes in projective space for general sets of places.

Findings

Zariski-density depends on the configuration of hyperplanes and the number field.

Complete characterization for the classical case with archimedean places.

Conditions under which integral points are not Zariski-dense.

Abstract

We study the S-integral points on the complement of a union of hyperplanes in projective space, where S is a finite set of places of a number field k. In the classical case where S consists of the set of archimedean places of k, we completely characterize, in terms of the hyperplanes and the field k, when the (S-)integral points are not Zariski-dense.