On \(\mathbb Q\)-derived polynomials.

*(English)*Zbl 1058.11045A 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.

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.

Reviewer: Bjorn Poonen (Berkeley)

##### MSC:

11G30 | Curves of arbitrary genus or genus \(\ne 1\) over global fields |