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Computing algebraic numbers of bounded height


Authors: John R. Doyle and David Krumm
Journal: Math. Comp. 84 (2015), 2867-2891
MSC (2010): Primary 11Y16; Secondary 11Y40
DOI: https://doi.org/10.1090/mcom/2954
Published electronically: April 1, 2015
MathSciNet review: 3378851
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Abstract: We describe an algorithm which, given a number field $ K$ and a bound $ B$, finds all the elements of $ K$ having relative height at most $ B$. Two lists of numbers are computed: one consisting of elements $ x\in K$ for which it is known with certainty that $ H_K(x)\leq B$, and one containing elements $ x$ such that $ \vert H_K(x)-B\vert<\theta $ for a tolerance $ \theta $ chosen by the user. We show that every element of $ K$ whose height is at most $ B$ must appear in one of the two lists.


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Additional Information

John R. Doyle
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: jdoyle@math.uga.edu

David Krumm
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: david.krumm@gmail.com

DOI: https://doi.org/10.1090/mcom/2954
Received by editor(s): November 18, 2011
Received by editor(s) in revised form: October 19, 2012, June 22, 2013, August 3, 2013, and February 25, 2014
Published electronically: April 1, 2015
Article copyright: © Copyright 2015 American Mathematical Society