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Algebraic integers whose conjugates all lie in an ellipse

Authors:	Valé rie Flammang Georges Rhin

Institution:	UMR CNRS 7122 Département de Mathématiques, UFR MIM, Université de Metz, Ile du Saulcy, 57045 Metz Cedex 01, France ; UMR CNRS 7122 Département de Mathématiques, UFR MIM, Université de Metz, Ile du Saulcy, 57045 Metz Cedex 01, France

Abstract:	We find all algebraic integers $\boldsymbol {\alpha }$ whose conjugates all lie in an ellipse with two of them nonreal, while the others lie in the real interval . 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 $\boldsymbol{\alpha}.$

Keywords:	Explicit auxiliary functions integer transfinite diameter

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