Counting points of fixed degree and bounded height

TL;DR

This paper derives asymptotic formulas for counting algebraic points of fixed degree and bounded height in projective space, expanding understanding of their distribution over number fields.

Contribution

It provides new asymptotic formulas for counting points of fixed degree and bounded height, including adelic-Lipschitz height, in projective space.

Findings

Asymptotic formula for points of degree e with bounded absolute height

Asymptotic formula for points with adelic-Lipschitz height

Extension of counting results to fixed degree points over number fields

Abstract

We consider the set of points in projective $n$-space that generate an extension of degree $e$ over given number field $k$, and deduce an asymptotic formula for the number of such points of absolute height at most $X$, as $X$ tends to infinity. We deduce a similar such formula with instead of the absolute height, a so-called adelic-Lipschitz height.