A note on the Casas-Alvero Conjecture

The Casas–Alvero conjecture predicts that every univariate polynomial $f$ over a field $K$ of characteristic zero having a common factor with each of its derivatives $H\mathrm{\_}i(f)$ is a power of a linear polynomial. Let $f=x^d+a\mathrm{\_}1x^{d-1}+\cdots +a\mathrm{\_}1x\in K[a\mathrm{\_}1,\dots ,a\mathrm{\_}d-1][x]$ and let $R\mathrm{\_}i=Res(f,H\mathrm{\_}i(f))\in K[a\mathrm{\_}1,\dots ,a\mathrm{\_}d-1]$ be the resultant of $f$ and $H\mathrm{\_}i(f)$, $i\in \{1,\dots ,d-1\}$. The Casas-Alvero Conjecture is equivalent to saying that $R\mathrm{\_}1,\dots ,R\mathrm{\_}d-1$ are ``independent’’ in a certain sense, namely that the height $ht(R\mathrm{\_}1,\dots ,R\mathrm{\_}d-1)=d-1$ in $K[a\mathrm{\_}1,\dots ,a\mathrm{\_}d-1]$. In this paper we prove a very partial result in this direction : if $i\in \{d-3,d-2,d-1\}$ then $R\mathrm{\_}i\mathrm{\notin }\sqrt{(R\mathrm{\_}1,\dots ,\breve{R\mathrm{\_}i},\dots ,R\mathrm{\_}d-1}$.