Casas–Alvero conjecture on common factors with Hasse derivatives

Let $K$ be a field, let $d$ be a positive integer, and let

$$f=x^d+a_1x^{d-1}+\cdots+a_{d-1}x+a_d\in K[x]$$

be a monic polynomial. For $i\in\{1,\ldots,d-1\}$, define its $i$-th Hasse derivative by

$$H_i(f)=\binom{d}{i}x^{d-i}+\binom{d-1}{i}a_1x^{d-i-1}+\cdots+\binom{i}{i}a_{d-i}.$$

Call $f$ a Casas–Alvero polynomial if, for every $i\in\{1,\ldots,d-1\}$, $f$ and $H_i(f)$ have a non-constant common factor. After translating a root in characteristic zero, assume $a_d=0$.

Casas–Alvero conjecture. If $\operatorname{char} K=0$ and $f\in K[x]$ is a Casas–Alvero polynomial of degree $d$ with $a_d=0$, then

$$f(x)=x^d.$$

Equivalently, after writing $R_i=\operatorname{Res}(f,H_i(f))$ in $K[a_1,\ldots,a_{d-1}]$ and $V=V(R_1,\ldots,R_{d-1})\subset K^{d-1}$, the conjecture says that $V=\{0\}$. This is a classical open problem; the paper proves a partial result for the final three resultants.