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

The house of an algebraic integer of degree d is the largest modulus of its conjugates. For d ≤ 28, we compute the smallest house > 1 of degree d, say m(d). As a consequence we improve Matveev’s theorem on the lower bound of m(d). We show that, in this range, the conjecture of SchinzelZassenhaus is satisfied. The minimal polynomial of any algebraic integer α whose house is equal to m(d) is a factor of a bi-, trior 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 C. They give better bounds than the classical ones for the coefficients of the minimal polynomial of an algebraic integer α whose house is small.