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Slicing the stars:
counting algebraic numbers, integers, and units by degree and
height
Robert Grizzard and Joseph Gunther |
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Vol. 11 (2017), No. 6, 1385–1436
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Abstract |
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Masser and Vaaler have given an asymptotic formula for the number of algebraic numbers of given degree and increasing height. This problem was solved by counting lattice points (which correspond to minimal polynomials over ) in a homogeneously expanding star body in . 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. |
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Keywords
arithmetic statistics, height, Mahler measure, geometry of
numbers
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Mathematical Subject Classification 2010
Primary: 11N45
Secondary: 11G50, 11H16, 11P21, 11R04, 11R06
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Milestones
Received: 6 December 2016
Revised: 16 March 2017
Accepted: 15 April 2017
Published: 16 August 2017
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Authors |
| Robert Grizzard | |
| Department of Mathematics University of Wisconsin-Madison 480 Lincoln Drive Madison, WI 53706 United States |
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| Joseph Gunther | |
| Department of Mathematics, The
Graduate Center City University of New York (CUNY) 365 Fifth Avenue New York, NY 10016 United States |
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