On the smallest houses of reciprocal algebraic integers

The house of an algebraic integer of degree d is the largest modulus of its conjugates. In this work, we compute the smallest houses mr(d) of all reciprocal algebraic integers for degree 28≤d≤42, and we suggest that mr(d)≥mr(20)20/d. In our computations, we use a family of explicit auxiliary function using the properties of reciprocal algebraic integer which give better bounds for Sk than the last one.