R.J. Stroeker
http://repub.eur.nl/ppl/8962/
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RePub, Erasmus University RepositoryOn Q-derived polynomials
http://repub.eur.nl/pub/71552/
Fri, 01 Dec 2006 00:00:01 GMT<div>R.J. Stroeker</div>
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. CopyrightComputing all integer solutions of a genus 1 equation
http://repub.eur.nl/pub/60849/
Wed, 01 Oct 2003 00:00:01 GMT<div>R.J. Stroeker</div><div>N. Tzanakis</div>
On Q-derived polynomials
http://repub.eur.nl/pub/553/
Wed, 18 Sep 2002 00:00:01 GMT<div>R.J. Stroeker</div>
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.Computing all integer solutions of a genus 1 equation
http://repub.eur.nl/pub/592/
Mon, 31 Dec 2001 00:00:01 GMT<div>R.J. Stroeker</div><div>N. Tzanakis</div>
The Elliptic Logarithm Method has been applied with great success
to the problem of computing all integer solutions of equations of
degree 3 and 4 defining elliptic curves. We extend this method
to include any equation f(u,v)=0 that defines a curve of genus 1.
Here f is a polynomial with integer coefficients and irreducible over
the algebraic closure of the rationals, but is otherwise of arbitrary shape and degree.
We give a detailed description of the general features of our approach,
and conclude with two rather unusual examples corresponding to equations
of degree 5 and degree 9.On Sums of Consecutive Squares
http://repub.eur.nl/pub/1355/
Sun, 01 Jan 1995 00:00:01 GMT<div>A. Bremner</div><div>R.J. Stroeker</div><div>N. Tzanakis</div>
In this paper we consider the problem of characterizing those perfect squares that can be expressed as the sum of consecutive squares where the initial term in this sum is the square of k. This problem is intimately related to that of finding all integral points on elliptic curves belonging to a certain family which can be represented by a Weierstrass equation with parameter k. All curves in this family have positive rank, and for those of rank 1 a most likely candidate generator of infinite order can be explicitly given in terms of k. We conjecture that this point indeed generates the free part of the Mordell-Weil
group, and give some heuristics to back this up. We also show that a point which is modulo torsion equal to a nontrivial multiple of this conjectured generator cannot be integral.
For k in the range 1...100 the corresponding curves are closely examined, all integral points are determined and all solutions to the original problem are listed. It is worth mentioning that all curves of equal rank in this family can be treated more or less uniformly in terms of the parameter k. The reason for this lies in the fact that in Sinnou David's lower bound of linear forms in elliptic logarithms - which is an essential ingredient of our approach - the rank is the dominant factor. Also the extra computational effort that is needed for some values of k in order to determine the rank unconditionally and construct a set of generators for the Mordell-Weil group deserves special attention, as there are some unusual features.On the equation Y2 = (X +p(X2 +p2)
http://repub.eur.nl/pub/68990/
Wed, 01 Jun 1994 00:00:01 GMT<div>R.J. Stroeker</div><div>J. Top</div>
In this paper the family of elliptic curves over Q given by the equation y2 = (x + p)(x2 + p2) is studied. It is shown that for p a prime number = ±3 mod 8, the only rational solution to the equation given here is the one with y = 0. The same is true for p = 2, Standard conjectures predict that the rank of the group of rational points is odd for all other primes p. A lot of numerical evidence in support of this is given. We show that the rank is bounded by 3 in general for prime numbers p. Moreover, this bound can only be attained for certain special prime numbers p = 1 mod 16. Examples of such rank 3 curves are given. Lastly, for certain primes p = 9 mod 16 nontrivial elements in the Shafarevich group of the elliptic curve arc constructed. In the literature one finds similar investigations of elliptic curves with complex multiplication. It may be interesting to note that the curves considered here do not admit complex multiplication. Copyright