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2017 Slicing the stars: counting algebraic numbers, integers, and units by degree and height
Robert Grizzard, Joseph Gunther
Algebra Number Theory 11(6): 1385-1436 (2017). DOI: 10.2140/ant.2017.11.1385

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 ) in a homogeneously expanding star body in 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.

Citation

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Robert Grizzard. Joseph Gunther. "Slicing the stars: counting algebraic numbers, integers, and units by degree and height." Algebra Number Theory 11 (6) 1385 - 1436, 2017. https://doi.org/10.2140/ant.2017.11.1385

Information

Received: 6 December 2016; Revised: 16 March 2017; Accepted: 15 April 2017; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 1375.11065
MathSciNet: MR3687101
Digital Object Identifier: 10.2140/ant.2017.11.1385

Subjects:
Primary: 11N45
Secondary: 11G50, 11H16, 11P21, 11R04, 11R06

Rights: Copyright © 2017 Mathematical Sciences Publishers

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Vol.11 • No. 6 • 2017
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