Details:
Title  On Qderived Polynomials  Author(s)  Roelof J. Stroeker  Text  To appear in Rocky Mountain Journal of Mathematics (2005)  Type  Technical Report, Misc  Abstract  A Qderived 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 Qderived 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 nonexistence 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 Qderived 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 Kderived polynomials of degree 4 with distinct zeros for itself and all its derivatives for a few real quadratic number fields K of small discriminant. 
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 Language  English  Year  2002  Edition  0  Translation 
No  Refereed 
No 
