The maximal modulus of an algebraic integer
David W. Boyd · Mathematics of Computation · 1985
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.