On the smallest value of the maximal modulus of an algebraic integer

Abstract:

The house of an algebraic integer of degree $d$ is the largest modulus of its conjugates. For $d\leq 28$, we compute the smallest house $>1$ of degree $d$, say $\text {m}(d)$. As a consequence we improve Matveev’s theorem on the lower bound of $\text {m}(d).$ We show that, in this range, the conjecture of Schinzel-Zassenhaus is satisfied. The minimal polynomial of any algebraic integer $\boldsymbol \alpha$ whose house is equal to $\text {m}(d)$ is a factor of a bi-, tri- or quadrinomial. The computations use a family of explicit auxiliary functions. These functions depend on generalizations of the integer transfinite diameter of some compact sets in $\mathbb {C}.$ They give better bounds than the classical ones for the coefficients of the minimal polynomial of an algebraic integer $\boldsymbol \alpha$ whose house is small.