2006-12-01
On Q-derived polynomials
Publication
Publication
Rocky Mountain Journal of Mathematics , Volume 36 - Issue 5 p. 1705- 1713
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. Copyright
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doi.org/10.1216/rmjm/1181069392, hdl.handle.net/1765/71552 | |
Rocky Mountain Journal of Mathematics | |
Organisation | Erasmus School of Economics |
Stroeker, R. (2006). On Q-derived polynomials. Rocky Mountain Journal of Mathematics, 36(5), 1705–1713. doi:10.1216/rmjm/1181069392 |