On Q-derived polynomials
AbstractA Q-derived polynomial is a univariate polynomial, defined overthe rationals, with the property that its zeros, and those of allits derivatives are rational numbers. There is a conjecture thatsays that Q-derived polynomials of degree 4 with distinctroots for themselves and all their derivatives do not exist. Weare not aware of a deeper reason for their non-existence than thefact that so far no such polynomials have been found. In thispaper an outline is given of a direct approach to the problem ofconstructing polynomials with such properties. Although noQ-derived polynomial of degree 4 with distinct zeros foritself and all its derivatives was discovered, in the process wecame across two infinite families of elliptic curves withinteresting properties. Moreover, we construct some K-derivedpolynomials of degree 4 with distinct zeros for itself and allits derivatives for a few real quadratic number fields K ofsmall discriminant.
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Bibliographic InfoPaper provided by Erasmus University Rotterdam, Econometric Institute in its series Econometric Institute Report with number EI 2002-30.
Date of creation: 18 Sep 2002
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Elliptic curve; Q-derived polynomial;
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