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On $$\mathbb Q$$-derived polynomials. (English) Zbl 1058.11045
A polynomial $$q(x) \in {\mathbb Q}[x]$$ of degree $$n$$ is called $$\mathbb Q$$-derived if all the zeros of $$q(x)$$ and its first $$n$$ derivatives lie in $$\mathbb Q$$. The article under review makes partial progress toward the diophantine problem of classifying such polynomials.
Say that a polynomial is of type $$p_{m_1,\dots,m_r}$$ if it has $$r$$ distinct roots, and $$m_i$$ is the multiplicity of the $$i^{\text{th}}$$ root. Say that $$q_1(x)$$ and $$q_2(x)$$ are equivalent if $$q_2(x)=rq_1(sx+t)$$ for some $$r,s,t \in {\mathbb Q}$$. R. H. Buchholz and J. A. MacDougall [J. Number Theory 81, 210–233 (2000; Zbl 1035.11009)] conjectured that (a) no polynomial of type $$p_{1,1,1,1}$$ is $${\mathbb Q}$$-derived, and (b) no polynomial of type $$p_{3,1,1}$$ is $${\mathbb Q}$$-derived. Assuming these two conjectures, they proved that all $$\mathbb Q$$-derived polynomials are equivalent to one of $$x^n$$, $$x^n(x-1)$$, $$x(x-1)\left(x - \frac{v(v-2)}{v^2-1} \right)$$, $$x^2(x-1)\left(x - \frac{9 (2w+z-12)(w+2)}{(z-w-18)(8w+z)} \right)$$ for some $$n \in {\mathbb Z}_{\geq 0}$$, $$v \in {\mathbb Q}$$, $$(w,z) \in E({\mathbb Q})$$ where $$E$$ is the elliptic curve $$z^2=w(w-6)(w+18)$$ (of rank 1).
The main result of the current article is to prove conjecture (b). The author shows that $$\mathbb Q$$-derived polynomials of type $$p_{3,1,1}$$ up to equivalence are parameterized by the rational points of a genus 5 curve $${\mathcal F}_1$$. The finite set $${\mathcal F}_1({\mathbb Q})$$ is calculated using methods similar to those in earlier papers of the author and others: The curve $${\mathcal F}_1$$ dominates a genus 2 curve $${\mathcal C}_1$$. The Mordell-Weil group of the jacobian of $${\mathcal C}_1$$ can be computed by $$2$$-descent, and this knowledge leads to a finite family of unramified covers of $${\mathcal C}_1$$ whose rational points together surject onto those of $${\mathcal C}_1$$. Only two of these covers have points over every completion of $$\mathbb Q$$. The rational points on these are calculated by the “elliptic Chabauty” method, which has been developed by N. Bruin [J. Reine Angew. Math. 562, 27–49 (2003; Zbl 1135.11320)] and by Flynn and J. R. Wetherell [Manuscr. Math. 100, 519–533 (1999; Zbl 1029.11024)].
The article also includes a discussion on finding rational points on curves of the form $$Y^2=(X^2-k)(X^2-rk)(X^2-r^2k)$$ with $$r,k \in {\mathbb Q}$$, $$k \not=0$$, $$r \not= 0,\pm 1$$; the curve $${\mathcal C}_1$$ is in this family. Finally, the author writes that conjecture (a) is out of reach of his methods, because the equivalence classes of $$\mathbb Q$$-derived polynomials of type $$p_{1,1,1,1}$$ are parameterized by a surface instead of a curve.

##### MSC:
 11G30 Curves of arbitrary genus or genus $$\ne 1$$ over global fields
##### Keywords:
genus 2; Jacobians; coverings; $$\mathbb Q$$-derived
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