Ideal-theoretic reformulation of the Casas–Alvero conjecture

Let $K$ be a field and let $f=x^d+a_1x^{d-1}+\cdots+a_{d-1}x$ be a monic degree-$d$ polynomial, where the constant term has been translated to zero. For $1\leq i\leq d-1$, let $H_i(f)$ be its $i$-th Hasse derivative and let

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

be the resultant of $f$ and $H_i(f)$. Set

$$V=V(R_1,\ldots,R_{d-1})\subset K^{d-1}.$$

Casas–Alvero ideal reformulation. The conjecture is equivalent to $V=\{0\}$, or, equivalently,

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

and hence to the existence of some $N\in\mathbb{N}$ such that $a_i^N\in(R_1,\ldots,R_{d-1})$ for every $i\in\{1,\ldots,d-1\}$. This is the resultant and algebraic-set formulation of the characteristic-zero Casas–Alvero conjecture: the common zero locus of the resultant conditions should consist only of the origin, corresponding to $f=x^d$.