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On \(\mathbb Q\)-derived polynomials. (English) Zbl 1140.11028

A \(\mathbb Q\)-derived polynomial is a polynomial \(f(x)\) with rational coefficients, with the property that all the roots of \(f(x)\) and of all the derivatives of \(f(x)\) are rational. A \(\mathbb Q\)-derived polynomial \(f(x)\) of degree \(n\) is said to be of type \(p_{1,\dots,1}\) (\(n\) 1’s) if all its zeros are distinct (i.e. no zero has multiplicity greater than one). Although it is not difficult to give examples of cubic \(\mathbb Q\)-derived polynomials which are of type \(p_{1,1,1}\) (e.g. \(f(x)=4x^3-4x^2-15x\)), it is not known yet whether there exists a \(\mathbb Q\)-derived quartic polynomial of type \(p_{1,1,1,1}\); and it is conjectured that no such polynomial exists.
The author shows that, in order to find a \(\mathbb Q\)-derived quartic polynomial of type \(p_{1,1,1,1}\), it suffices to find rational points \(P_1(t)\), \(P_2(t)\) with the same abscissa on two explicitly stated elliptic curves \(E_1(t)\), \(E_2(t)\) (\(t\) is a rational parameter) and then, by mean of these common abscissas construct quartic \(\mathbb Q\)-derived polynomials which, however, will not necessarily be of type \(p_{1,1,1,1}\). The author first studies a few properties of the elliptic curves \(E_1(t)\), \(E_2(t)\). Then, his search for two points \(P_1(t)\), \(P_2(t)\) as above, is reduced to finding rational points on certain hyperelliptic curves. In his examples he makes use of the computer package ratpoints (developed by C. Stahlke and M. Stoll) which searches in a clever way rational points on hyperelliptic curves. All the rational points found give rise to quartic \(\mathbb Q\)-derived polynomials with multiple roots. Thus, the new approach proposed in this paper results to supporting the above mentioned conjecture about non-existence of quartic \(\mathbb Q\)-derived polynomials of type \(p_{1,1,1,1}\). It is worth-noticing, however, that, with his method, the author is able to furnish quartic \(K\)-derived polynomial, where \(K\) is a quadratic number-field.

MSC:

11G05 Elliptic curves over global fields
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11C08 Polynomials in number theory

Software:

APECS; mwrank
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] R.H. Buchholz and J.A. MacDougall, When Newton met Diophantus : A study of rational-derived polynomials and their extension to quadratic fields , J. Number Theory 81 (2000), 210-233. · Zbl 1035.11009 · doi:10.1006/jnth.1999.2473
[2] I. Connell’s, Apecs -\(6.1\) is available at the web site http://www.math.mcgill.ca in the directory /connell/public/apecs/. · JFM 59.0077.06
[3] J. Cremona’s, mwrank is available at the web site http://www.maths.nott.ac.uk in the directory /personal/jec/ftp/progs/. · Zbl 1246.01050
[4] E.V. Flynn, On \(\Q\)-derived polynomials , Proc. Edinburgh Math. Soc. 44 (2001), 103-110. · Zbl 1058.11045 · doi:10.1017/S0013091599000760
[5] M. Stoll, The program, ratpoints is based on an idea of Noam Elkies, with many improvements by Colin Stahlke and Michael Stoll (version 1.5, 23 April 2001). · Zbl 0987.31007
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