The maximal modulus of an algebraic integer

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 ⁡ θ 0 \log {\theta _0} , where θ 0 {\theta _0} is the smallest P.V. number. They also shed some light on a recent conjecture of Lind concerning the Perron numbers.