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 $H\_i(f)$ is a power of a linear polynomial. Let $f=x^d+a\_1x^{d-1}+\cdots+a\_1x \in K[a\_1,\ldots,a\_{d-1}][x]$ and let $R\_i = Res(f,H\_i(f))\in K[a\_1,\ldots,a\_{d-1}]$ be the resultant of $f$ and $H\_i(f)$, $i \in \{1,\ldots,d-1\}$. The Casas-Alvero Conjecture is equivalent to saying that $R\_1,\ldots,R\_{d-1}$ are ``independent'' in a certain sense, namely that the height $ht(R\_1,\ldots,R\_{d-1})=d-1$ in $K[a\_1,\ldots,a\_{d-1}]$. In this paper we prove a very partial result in this direction : if $i \in \{d-3,d-2,d-1\}$ then $R\_i \notin \sqrt{(R\_1,\ldots,\breve{R\_i},\ldots,R\_{d-1}}$.