Vol. 11, No. 6, 2017

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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

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.

arithmetic statistics, height, Mahler measure, geometry of numbers
Mathematical Subject Classification 2010
Primary: 11N45
Secondary: 11G50, 11H16, 11P21, 11R04, 11R06
Received: 6 December 2016
Revised: 16 March 2017
Accepted: 15 April 2017
Published: 16 August 2017
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