The maximal modulus of an algebraic integer
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- by David W. Boyd PDF
- Math. Comp. 45 (1985), 243-249 Request permission
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.References
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Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Math. Comp. 45 (1985), 243-249
- MSC: Primary 11R06
- DOI: https://doi.org/10.1090/S0025-5718-1985-0790657-8
- MathSciNet review: 790657