SIMON-MARAIS 2021 B4

The following problem is open in the sense that the answer to part (b) is not currently known. A proof of part (a) will be awarded 5 points. Up to 7 additional points may be awarded for progress on part (b).

Let $p(x)$ be a polynomial of degree $d$ with coefficients belonging to the set of rational numbers $\mathbb{Q}$ . Suppose that, for each $1 \le k \le d-1$ , $p(x)$ and its $k$ th derivative $p^{(k)}(x)$ have a common root in $\mathbb{Q}$ ; that is, there exists $r_k \in \mathbb{Q}$ such that $p(r_k) = p^{(k)}(r_k) = 0$ . (a) Prove that if $d$ is prime then there exist constants $a, b, c \in \mathbb{Q}$ such that $$p(x) = c(ax + b)^d.$$ (b) For which integers $d \ge 2$ does the conclusion of part (a) hold?