Integral points of fixed degree and bounded height

TL;DR

This paper investigates the asymptotic behavior of algebraic integer points of fixed degree and bounded height in affine space, extending previous results to the case where coordinates are algebraic integers.

Contribution

It derives new asymptotic formulas for the count of algebraic integer points of fixed degree and bounded height, generalizing earlier results for all algebraic points.

Findings

Asymptotic formulas for algebraic integer points of fixed degree

Extension of previous results to algebraic integers

Quantitative bounds on the number of such points

Abstract

By Northcott's Theorem there are only finitely many algebraic points in affine $n$-space of fixed degree over a given number field and of height at most $X$. For large $X$ the asymptotics of these cardinalities have been investigated by Schanuel, Schmidt, Gao, Masser and Vaaler, and the author. In this paper we study the case where the coordinates of the points are restricted to algebraic integers, and we derive the analogues of Schanuel's, Schmidt's, Gao's and the author's results.