A Q-derived polynomial is a univariate polynomial, defined overthe rationals, with the property that its zeros, and those of allits derivatives are rational numbers. There is a conjecture thatsays that Q-derived polynomials of degree 4 with distinctroots for themselves and all their derivatives do not exist. Weare not aware of a deeper reason for their non-existence than thefact that so far no such polynomials have been found. In thispaper an outline is given of a direct approach to the problem ofconstructing polynomials with such properties. Although noQ-derived polynomial of degree 4 with distinct zeros foritself and all its derivatives was discovered, in the process wecame across two infinite families of elliptic curves withinteresting properties. Moreover, we construct some K-derivedpolynomials of degree 4 with distinct zeros for itself and allits derivatives for a few real quadratic number fields K ofsmall discriminant.Elliptic curve;Q-derived polynomial

Stroeker, R.J.

English

Research Papers in Economics

On Q-derived Polynomials
R.J. Stroeker¤
Econometric Institute, EUR
Report EI 2002-30
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.
2000 Mathematics subject classiﬁcation: 11G30
Key words and phrases: Q-derived polynomial, elliptic curve
¤Econometric Institute, Erasmus University, P.O.Box 1738, 3000 DR Rotterdam, The Netherlands;
e-mail: stroeker@few.eur.nl; homepage: http://www.few.eur.nl/few/people/stroeker/On Q-derived Polynomials 2
1 Introduction
A polynomial f 2 Q[x] of degree n with n rational zeros and such that all of its derivatives
of positive degree have the same property is known as a Q-derived polynomial. It is easy
to classify all such polynomials of degree at most 3. For instance,
f(x) = 4x
3 ¡ 4x
2 ¡ 15x = x(2x + 3)(2x ¡ 5)
f
0(x) = 12x
2 ¡ 8x ¡ 15 = (2x ¡ 3)(6x + 5)
f
00(x) = 24x ¡ 8
is a Q-derived polynomial of type p1;1;1. By deﬁnition, the class pm1;:::;ms contains all
polynomials with s distinct zeros, where the i-th zero has multiplicity mi (i = 1;:::;s)
and
Ps
i=1 mi = n = deg(f). Surprisingly, polynomials of type p1;1;1;1 are unknown. More
strongly, it is conjectured that such polynomials do not exist (see [1] and [4]).
Conjecture 1.1. There is no Q-derived polynomial of type p1;1;1;1.
This conjecture is the only remaining obstacle in the course of a full classiﬁcation of all
quartic Q-derived polynomials (see [4]). In the closing lines of [4] it is observed that
this paper’s approach to proving the non-existence of Q-derived polynomials of type p3;1;1
won’t work for the p1;1;1;1 case. Therefore, in the next section we shall outline a diﬀerent,
constructive way of attacking this remaining conjecture.
2 Q-derived polynomials of degree 4
Clearly, if f(x) is Q-derived, then so is g(x) = r1f(r2(x + r3)) for any ri 2 Q (i = 1;2;3),
r2 6= 0. This observation implies that without loss of generality we may take f0(0) = 0,
f(1) = 0, and f to be monic for suitable choices of r3, r2, r1 respectively, provided f is
Q-derived with at least two diﬀerent zeros. From now on we assume this to be the case.
Let
f(x) = (x ¡ 1)(x ¡ a)(x ¡ b)(x ¡ c) (1)
be Q-derived with a;b;c 2 Q and f0(0) = 0, which yields
abc + ab + bc + ca = 0 (2)
Further f is Q-derived and so f0(x) = 0 and f00(x) = 0 have rational roots only. Because
f0(0) = 0, this yields two quadratic polynomials with rational discriminants:
9(a + b + c + 1)
2 ¡ 32(ab + bc + ca + a + b + c) = rational square (3)
9(a + b + c + 1)
2 ¡ 24(ab + bc + ca + a + b + c) = rational square (4)
Now we shall assume that bc + b + c 6= 0, for otherwise it follows from (2) that b = c = 0
and f cannot be of type p1;1;1;1. These f’s are adequately dealt with in [1]. For the sameOn Q-derived Polynomials 3
reason we assume a 6= 0, b 6= 0, and c 6= 0.
Next we eliminate a from (2) and each of the equations in (3) separately. Working with
Maple, this can be conveniently done using Groebner bases with total degree order (a;b;c)
for each pair. Setting b = 1=t and c = (U ¡ 1)=(t + 1) and adjusting for rational squares
gives two similar parametric families of elliptic curves with correspondig quartic equations
E1(t) : V
2 = U4 ¡ (14
3 T + 2)U3 + (9T 2 + 14
3 T ¡ 5
9)U2 ¡ (14
3 T + 2)U + 1 (5)
E2(t) : W
2= U4 ¡ 2(T + 1)U3 + (9T 2 + 2T + 1
3)U2 ¡ 2(T + 1)U + 1 (6)
where we write T = (t2 + t + 1)=3t for convenience. These curves have to be investigated
for rational points with common abscissa. We have to keep in mind however that not all
pairs of points with common abscissa qualify. For instance, the points (U;V ) = (0;§1) and
(U;W) = (0;§1) give rise to bc+b+c = 0, which we excluded. This can be seen as follows.
If U is the common abscissa of two points, one on each of the curves E1(t) and E2(t) (see
(5) and (6)), then combining b = 1=t with c = (U ¡1)=(t+1) gives bc+b+c = U=t, which
vanishes with U. Observe that for U 6= 0 we have
a = ¡
U ¡ 1
U(t + 1)
; b =
1
t
; c =
U ¡ 1
t + 1
(7)
There are many solutions to the 4-th degree Q-derived problem with multiple zeros. Cor-
responding points for such solutions may be obtained by equating any two elements of the
set f1;a;b;cg. Proceeding in this way1 we obtained the following U-values for these points.
U =
1
t + 2
(a = 1); t + 2 (c = 1);
t
2t + 1
(a = b);
2t + 1
t
(b = c); ¡1 (c = a) (8)
Further, two or more of these U values can only be equal for t 2 f1;¡1;¡3;¡1
3g. Con-
sequently, we may expect these families of elliptic curves to have many rational points,
which suggests a rank of at least 1 for all rational values of t, with a few obvious excep-
tions. What we hope to ﬁnd is an explicit rational point P(t) with abscissa U(t), which
is a point on one of these two curves for all t, and which corresponds to a point Q(t) on
the other curve that can be forced to have the same abscissa U(t) for a suitably chosen
rational value t. If such a point exists, and if it does not correspond to a solution with
multiple zeros, then we have found a Q-derived polynomial of degree 4 and of type p1;1;1;1.
So we have to investigate the properties of these two families of elliptic curves.
3 Two Families of Elliptic Curves
The curve E1(t)=Q has the Weierstraß equation
Y
2 ¡ 2(7T + 3)XY ¡ 36(7T + 3)Y = X
3 + 2(16T
2 ¡ 7)X
2 ¡ 324X ¡ 648(16T
2 ¡ 7) (9)
1The choice b = 1 leads to t = 1 and the curve E1(1) is not elliptic.On Q-derived Polynomials 4
and a Weierstraß equation for E2(t)=Q is
Y
2 ¡ 2(3T + 1)XY ¡ 36(3T + 1)Y = X
3 + 2(36T
2 ¡ 3)X
2 ¡ 324X ¡ 648(36T
2 ¡ 3) (10)
Recall that T = (t2 + t + 1)=3t. The discriminant of E1(t)=Q is
2
183
4(T ¡ 1)
2(T + 1)
2(9T + 7)
2(3T + 1)(27T ¡ 23)
It readily follows that E1(t) is non-singular and hence elliptic for t 62 f0;1;¡1;¡3;¡1
3g.
Further, for transcendental T we have E1(Q(T))tor » = Z2 with non-trivial torsion point
P0(t) with coordinates (X;Y ) = (¡18;0) on the cubic (9).
Considering U = ¡1;t;t+2 (see (8)) and the point producing maps t 7! 1=t and (U;V ) 7!
(1=U;V=U2), we ﬁnd a collection of points belonging to the subgroup of E1(t)(Q) generated
by two points of inﬁnite order, say P1(t) with coordinates (U;V ) = (¡1;3T + 7
3) on the
quartic (5) and (X;Y ) = (96T + 78;¡192T 2 ¡ 576T ¡ 384) on the cubic (9), and P2(t)
with coordinates (U;V ) = (2 + 1=t;¡2(t + 1)(t ¡ 1)(3t + 1)=3t2) on (5) and (X;Y ) =
(¡16t ¡ 2;32(t ¡ 1)(t2 ¡ 2t ¡ 2)=3t) on (9). These points need not be independent for all
t, but often they are. Therefore we have
Theorem 3.1. The curve E1(t)=Q is elliptic for all t 62 f0;1;¡1;¡3;¡1
3g. For these
values of t its torsion group has a subgroup of order 2 and its rank is at least 1.
In fact, E1(t) can have three points of order 2, a situation that can be parameterized as
follows. Setting Y = (7T + 3)X + 18(7T + 3) in (9) leads to
(X + 18)(X
2 + (81T
2 + 42T ¡ 23)X + 306T
2 + 756T + 414 = 0
For the second factor to be rationally solvable, its discriminant needs to be a perfect square,
which means
(3T + 1)(27T ¡ 23)(9T + 7)
2 = rational square
As none of the factors in the left-hand side can vanish, and replacing T by (t2 +t+1)=3t,
we ﬁnd that (9t ¡ 7)2 + 32 = rational square. This ﬁnally yields the parameterization
1 + t
1 ¡ t
= u ¡
2
u
; for u 2 Q but u 6= 0;§1;§2
The lower bound 1 for the rank is best possible as can be seen from the tabel below.
The curve E2(t)=Q has discriminant
2
143
6(3T
2 ¡ 1)
2(3T ¡ 1)(27T ¡ 1)(8T
2 + 42T + 25)
and we check that E2(t) is non-singular for all t 6= 0. Again, for transcendental T we have
E2(Q(T))tor » = Z2 with non-trivial torsion point (X;Y ) = (¡18;0) on the cubic (10). We
have
Theorem 3.2. The curve E2(t)=Q is elliptic for all t 6= 0. For t 6= 0 its torsion group has
a subgroup of order 2 and its rank is at least 1.On Q-derived Polynomials 5
The point P = (X;Y ) = (18;0) is on both curves (9) and (10). This point cannot be of
ﬁnite order. On the curve (9) we have ¡P = (18;72(7T + 3)) and 7T + 3 6= 0, so that
P cannot be of order 2. More tedious arguments, using Maple and Apecs, show that P
cannot be of order 3;4;5;6;7;8;9;10 or 12 either. For instance, setting x[2P] = ¡x[P]
gives T = ¡2 or T = ¡5=8, both of which are impossible, so that P cannot be of order 3.
And so on. In the same way we check this for the second curve.
In the table below we have gathered information on the curves E1(t) and E2(t) for all
125 values of t in Q \ [¡1;1] of height at most 20 and with t 6= 0;¡1;1;¡3;¡1
3. The
computations were carried out with Ian Connell’s Maple package Apecs 6 [2]. We also ran
John Cremona’s mwrank [3] under Linux to verify Apecs’ results. For only three values of
t, namely t = ¡2=11;¡5=8;4=5, we were unable to ﬁnd a guaranteed value for the rank of
E1(t)=Q. The best we can do in this respect is to say that the ranks of these three curves
are 1, 2, or 3.
Torsion group Rank
Z2 Z2 £ Z2 1 2 3 4
E1 120 5 3 80 38 4
E2 125 0 ¤39 63 18 5
Table 1: Number of curves Ei(t) (i = 1;2) for 125 values of t. The ¤ means that three curves
counted here may have rank 2 or 3, instead of 1.
4 Finding Suitable Rational Points
As we explained at the end of section 2, we approach the problem of ﬁnding a Q-derived
polynomial of type p1;1;1;1 by selecting a point on E1(t) (5) for general t, and trying to
force its abscissa U(t) to be the abscissa of a point on E2(t) (6) for a suitable choice of t.
This works as follows. We take any linear combination of P0(t), P1(t) and P2(t) (see the
previous section), extract its coordinate U(t) and substitute this into (6), which then can
be written as F(t) = Z2 for Z 2 Q(t). Next we try to ﬁnd a suitable rational point on
this elliptic or generally hyperelliptic equation. This search is much simpliﬁed by the very
fast program ratpoints of Michael Stoll [5], which we need to extend to cover deg(F) > 10.
In fact we did our calculations with all 127 points P(t) = c0P0(t) + c1P1(t) + c2P2(t) for
integers ci with c0 = 0;1 and ci (i = 1;2) between ¡3 and 4, and with ratpoints searching
up to height 100000. The highest degree of F we came across was 76. Unfortunately, we
did not ﬁnd any p1;1;1;1-type polynomial. We give a few examples.On Q-derived Polynomials 6
4.1 Example
For P(t) = P0(t) + P1(t) + P2(t) we have U(t) = 1=(t + 2), which corresponds to a = 1.
Substitution of U(t) = 1=(t + 2) into (6) yields
F1(t) := 3(t
4 + 4t
3 + 4t
2 + 3) =
µ
3t(t + 2)2
2(t + 1)
W
¶2
The elliptic curve Z2 = F1(t) has rank 2, but every rational point of this curve corresponds
to a Q-derived polynomial with a double root 1.
4.2 Example
Take P(t) = P0(t) + 3P1(t) ¡ P2(t). Then U(t) = (5t + 4)=(2t2 + 2t ¡ 1) and substitution
of this U(t) into (6) results in
F2(t) := 3(67t
8+228t
7+306t
6¡92t
5¡63t
4+276t
3+214t
2+24t+12) =
µ
3t(2t2 + 2t ¡ 1)2
2(t + 1)
W
¶2
Searching for rational points on the curve Z2 = F2(t) with ratpoints-1.5 we only ﬁnd the
ﬁnite rational points corresponding to t = 1;¡1;¡3;¡4
5, all four values giving rise to
inadmissible situations, the last one notably U = 0.
Observe that for rational t we generally get a K-derived polynomial of type p1;1;1;1 over the
quadratic ﬁeld K. For instance, setting t = ¡2 ﬁnds such a polynomial over K = Q(
p
3).
The roots of this polynomial are [a;b;c;d] = [3
2;¡1
2;3;1].
5 K-derived polynomials for some number ﬁelds K
In this section we give some examples of K-derived polynomials of type p1;1;1;1 over the
number ﬁeld K of small discriminant. Obviously, for a suitable U(t), any rational t will
generally give a K-derived polynomial of type p1;1;1;1 over a quadratic number ﬁeld K.
There is however no reason for the discriminant of K to be small. We have already given
an example of such a polynomial over K = Q(
p
3) in Example 4.2.
5.1 Example
Setting U(t) = t in (6) gives
W
2 =
4
3
(t
2 + 1)
2
so that for each t 6= 0;1;¡1 the polynomial with roots (7)
[a;b;c;d] = [¡
t ¡ 1
t(t + 1)
;
1
t
;
t ¡ 1
t + 1
;1]
is a K-derived polynomial of type p1;1;1;1 over K = Q(
p
3).On Q-derived Polynomials 7
5.2 Example
The U-value of P(t) = P0(t) ¡ P1(t) + P2(t) is
U(t) =
2t2 + 2t ¡ 1
5t + 4
Let K = Q(
p
D) for squarefree D. The following t-values give diﬀerent K-derived polyno-
mials of type p1;1;1;1 for the relevant D as shown in the table below.
D t
3 1
7, 2
3
3 ¢ 31 ¢ 37 ¡ 7
17, 1
4, 1
2, 4
37 ¢ 103 ¡2
7, 3
9931 ¡3
7, 10
Table 2: Some K-derived polynomials of type p1;1;1;1 over a quadratic number ﬁeld K = Q(
p
D)
of small discriminant.
References
[1] R.H. Buchholz & J.A. MacDougall, When Newton met Diophantus: A Study
of rational-derived polynomials and their extension to quadratic ﬁelds, J. Number Th.
81(2) (2000), 210–233.
[2] I. Connell’s Apecs-6:1 is available at the web site http://www.math.mcgill.ca in the
directory /connell/public/apecs/.
[3] J. Cremona’s mwrank is available at the website http://www.maths.nott.ac.uk in
the directory /personal/jec/ftp/progs/.
[4] E.V. Flynn, On Q-Derived Polynomials, Proc. Edinburgh Math. Soc. 44 (2001),
103–110.
[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).

Connell’s Apecs-6:1 is available at the web site http://www.math.mcgill.ca in the directory /connell/public/apecs/.

On Q-derived polynomials

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