Details:
Title | On Q-derived Polynomials | Author(s) | Roelof J. Stroeker | Text | To appear in Rocky Mountain Journal of Mathematics (2005) | Type | Technical Report, Misc | 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. |
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| Language | English | Year | 2002 | Edition | 0 | Translation |
No | Refereed |
No |
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