#### Vol. 11, No. 6, 2017

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

### Robert Grizzard and Joseph Gunther

Vol. 11 (2017), No. 6, 1385–1436
##### 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 $ℤ$) in a homogeneously expanding star body in ${ℝ}^{d+1}\phantom{\rule{0.3em}{0ex}}$. 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.

##### Keywords
arithmetic statistics, height, Mahler measure, geometry of numbers
##### Mathematical Subject Classification 2010
Primary: 11N45
Secondary: 11G50, 11H16, 11P21, 11R04, 11R06