Algebra & Number Theory

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

Robert Grizzard and 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 ) 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.

Article information

Source
Algebra Number Theory, Volume 11, Number 6 (2017), 1385-1436.

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

Permanent link to this document
https://projecteuclid.org/euclid.ant/1510842805

Digital Object Identifier
doi:10.2140/ant.2017.11.1385

Mathematical Reviews number (MathSciNet)
MR3687101

Zentralblatt MATH identifier
1375.11065

Subjects
Primary: 11N45: Asymptotic results on counting functions for algebraic and topological structures
Secondary: 11G50: Heights [See also 14G40, 37P30] 11H16: Nonconvex bodies 11P21: Lattice points in specified regions 11R04: Algebraic numbers; rings of algebraic integers 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure

Keywords
arithmetic statistics height Mahler measure geometry of numbers

Citation

Grizzard, Robert; Gunther, Joseph. Slicing the stars: counting algebraic numbers, integers, and units by degree and height. Algebra Number Theory 11 (2017), no. 6, 1385--1436. doi:10.2140/ant.2017.11.1385. https://projecteuclid.org/euclid.ant/1510842805


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