On Families of Irreducible Polynomials With Constraints on Height, Mahler Measure, and Graeffe Transforms

A family of polynomials refers to an infinite collection of polynomials with integer coefficients, typically defined in a parametric form. Constructing such a family in which all elements are irreducible is a major challenge, especially when constraints are imposed both on their coefficients and on the location of their complex roots. In this article, we present explicit families of irreducible polynomials of height equal to 1, whose Mahler measure is strictly greater than 1, and whose Graeffe-Dandelin transforms have prescribed heights. More precisely, we construct a family of irreducible polynomials that provides an affirmative answer to a question recently raised in the literature: does there exist an infinite number of irreducible polynomials P, of height equal to 1, such that \M(P) > 1, \delta(P) = 1, and the Graeffe transform \cG P also has height 1?