Slicing the stars: counting algebraic numbers, integers, and units by degree and height

TL;DR

This paper develops advanced lattice point counting techniques to asymptotically count algebraic numbers, integers, and units by degree and height, refining previous volume estimates and error bounds for higher-codimension slices.

Contribution

It introduces methods to estimate volumes of higher-codimension slices in star bodies, enabling counting of algebraic units and integers with explicit error terms.

Findings

Derived explicit asymptotic formulas for counting algebraic numbers and units.

Improved error bounds in lattice point counting with better power savings.

Extended volume calculations to higher-codimension slices for algebraic number counting.

Abstract

Masser and Vaaler have given an asymptotic formula for the number of algebraic numbers of given degree $d$ and increasing height. This problem was solved by counting lattice points (which correspond to minimal polynomials over $\mathbb{Z}$) in a homogeneously expanding star body in $\mathbb{R}^{d+1}$. The volume of this star body was computed by Chern and Vaaler, who also computed the volume of the codimension-one "slice" corresponding to monic polynomials -- this led to results of Barroero on counting algebraic integers. We show how to estimate the volume of higher-codimension slices, which allows us to count units, algebraic integers of given norm, trace, norm and trace, and more. We also refine the lattice point-counting arguments of Chern-Vaaler to obtain explicit error terms with better power savings, which lead to explicit versions of some results of Masser-Vaaler and Barroero.