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

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 ${ℝ}^{d+1}$. 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.