Template-Type: ReDIF-Paper 1.0
Author-Name: Stroeker, R.J.
Author-Name-Last: Stroeker
Author-Name-First: Roel
Title: On Q-derived polynomials
Abstract: A Q-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 Q-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 Q-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 K-derived polynomials of degree 4 with distinct zeros for itself and all its derivatives for a few real quadratic number fields K of small discriminant.
Creation-Date: 2002-09-18
File-URL: https://repub.eur.nl/pub/553/feweco20020918141138.pdf
File-Format: application/pdf
Series: RePEc:ems:eureir
Number: EI 2002-30
Keywords: Elliptic curve, Q-derived polynomial
Handle: RePEc:ems:eureir:553