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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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

arithmetic statistics height Mahler measure geometry of numbers


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.

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