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

Grizzard, Robert; Gunther, Joseph

doi:10.2140/ant.2017.11.1385

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

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

Robert Grizzard and Joseph Gunther

Vol. 11 (2017), No. 6, 1385–1436

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 $ℤ$ ) in a homogeneously expanding star body in $ℝ^{d + 1}$ . 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.

Keywords

arithmetic statistics, height, Mahler measure, geometry of numbers

Mathematical Subject Classification 2010

Primary: 11N45

Secondary: 11G50, 11H16, 11P21, 11R04, 11R06

Milestones

Received: 6 December 2016

Revised: 16 March 2017

Accepted: 15 April 2017

Published: 16 August 2017

Authors

Robert Grizzard
Department of Mathematics University of Wisconsin-Madison 480 Lincoln Drive Madison, WI 53706 United States

Joseph Gunther
Department of Mathematics, The Graduate Center City University of New York (CUNY) 365 Fifth Avenue New York, NY 10016 United States

Vol. 11, No. 6, 2017