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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Algebraic integers whose conjugates all lie in an ellipse


Authors: Valérie Flammang and Georges Rhin
Journal: Math. Comp. 74 (2005), 2007-2015
MSC (2000): Primary 11R04, 11Y40, 12D10
Published electronically: March 8, 2005
MathSciNet review: 2164108
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Abstract: We find all $15909$ algebraic integers $\boldsymbol {\alpha }$ whose conjugates all lie in an ellipse with two of them nonreal, while the others lie in the real interval $[-1,2]$. This problem has applications to finding certain subgroups of $SL(2,\mathbb{C})$. We use explicit auxiliary functions related to the generalized integer transfinite diameter of compact subsets of $\mathbb{C}$. This gives good bounds for the coefficients of the minimal polynomial of $\boldsymbol{\alpha}.$


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Additional Information

Valérie Flammang
Affiliation: UMR CNRS 7122 Département de Mathématiques, UFR MIM, Université de Metz, Ile du Saulcy, 57045 Metz Cedex 01, France
Email: flammang@poncelet.univ-metz.fr

Georges Rhin
Affiliation: UMR CNRS 7122 Département de Mathématiques, UFR MIM, Université de Metz, Ile du Saulcy, 57045 Metz Cedex 01, France
Email: rhin@poncelet.univ-metz.fr

DOI: http://dx.doi.org/10.1090/S0025-5718-05-01735-7
PII: S 0025-5718(05)01735-7
Keywords: Explicit auxiliary functions, integer transfinite diameter.
Received by editor(s): December 19, 2003
Received by editor(s) in revised form: May 13, 2004
Published electronically: March 8, 2005
Article copyright: © Copyright 2005 American Mathematical Society