A note on the Casas-Alvero Conjecture

Abstract

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