Algebraic integers whose conjugates all lie in an ellipse

Flammang, Valérie; Rhin, Georges

doi:10.1090/S0025-5718-05-01735-7

Algebraic integers whose conjugates all lie in an ellipse
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by Valérie Flammang and Georges Rhin;

Math. Comp. 74 (2005), 2007-2015

DOI: https://doi.org/10.1090/S0025-5718-05-01735-7

Published electronically: March 8, 2005

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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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