On $\Q$-Derived Polynomials

Abstract

A -derived polynomial is a univariate polynomial, defined over the rationals, with the property that its zeros, and those of all its derivatives are rational numbers. There is a conjecture that says that -derived polynomials of degree 4 with distinct roots for themselves and all their derivatives do not exist. We are not aware of a deeper reason for their non-existence than the fact that so far no such polynomials have been found. In this paper an outline is given of a direct approach to the problem of constructing polynomials with such properties. Although no -derived polynomial of degree 4 with distinct zeros for itself and all its derivatives was discovered, in the process we came across two infinite families of elliptic curves with interesting properties. Moreover, we construct some -derived polynomials of degree 4 with distinct zeros for itself and all its derivatives for a few real quadratic number fields of small discriminant.