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arXiv · 1609.08720

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

Robert Grizzard·Joseph Gunther
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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.

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Robert Grizzard, Joseph Gunther. 2016-09-28. Slicing the stars: counting algebraic numbers, integers, and units by degree and height. https://doi.org/10.2140/ant.2017.11.1385

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