Radical ideal formulation of the Casas–Alvero conjecture

Let $K$ be an algebraically closed field of characteristic zero, let $d$ be a positive integer, and let $f=x^d+a_1x^{d-1}+\cdots+a_{d-1}x\in K[a_1,\ldots,a_{d-1}][x]$. For $i\in\{1,\ldots,d-1\}$, let

$$R_i=\operatorname{Res}(f,H_i(f))\in K[a_1,\ldots,a_{d-1}],$$

where $H_i(f)$ is the $i$-th Hasse derivative of $f$.

Nullstellensatz formulation of the Casas–Alvero conjecture. The conjecture is equivalent to

$$\sqrt{(R_1,\ldots,R_{d-1})}=(a_1,\ldots,a_{d-1}),$$

or, equivalently, there is some $N\in\mathbb{N}$ such that

$$a_i^N\in(R_1,\ldots,R_{d-1})\quad\text{for all }i\in\{1,\ldots,d-1\}.$$

This is an algebraic reformulation of the preceding Casas–Alvero conjecture rather than a separate mathematical claim. Over non-algebraically closed fields, the source notes that this formulation is a priori stronger than the corresponding statement about the $K$-rational zero set.