The Reciprocal Algebraic Integers Having Small House

Abstract

Let α be a reciprocal algebraic integer of degree d. The house of α is the largest modulus of its conjugates. We compute the minimum of the houses of all reciprocal algebraic integers of degree d having the minimal polynomial which is a factor of a Dth degree reciprocal or antireciprocal polynomial with at most eight monomials, say , for d at most 180, and . We show that it is not necessary to take into account imprimitive polynomials. The computations suggest several conjectures. We show that d-th power of the house of a sequence of reciprocal primitive polynomials has a limit. We present a property of antireciprocal hexanomials.