## Algebra & Number Theory

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

#### 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 $ℝd+1$. 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
Revised: 16 March 2017
Accepted: 15 April 2017
First available in Project Euclid: 16 November 2017

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

#### 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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