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The maximal modulus of an algebraic integer

Author: David W. Boyd
Journal: Math. Comp. 45 (1985), 243-249, S17
MSC: Primary 11R06
MathSciNet review: 790657
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Abstract: The maximal modulus of an algebraic integer is the absolute value of its largest conjugate. We compute the minimum of the maximal modulus of all algebraic integers of degree d which are not roots of unity, for d at most 12. The computations suggest that the minimum is never attained for a reciprocal algebraic integer. The truth of this conjecture would show that the conjecture of Schinzel and Zassenhaus follows from a theorem of Smyth. We further test our conjecture by computing the minimum of the maximal modulus of all reciprocal algebraic integers of degree d which are not roots of unity, for d at most 16. Our computations strongly suggest that the best constant in the conjecture of Schinzel and Zassenhaus is 1.5 $ \log {\theta _0}$, where $ {\theta _0}$ is the smallest P.V. number. They also shed some light on a recent conjecture of Lind concerning the Perron numbers.

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Keywords: Algebraic integer, maximal modulus, Schinzel-Zassenhaus conjecture, Perron numbers, Smyth's theorem, Newton's formulas
Article copyright: © Copyright 1985 American Mathematical Society

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