  The Casas-Alvero conjecture predicts that every univariate polynomial over a
field of characteristic zero having a common factor with each of its
derivatives $H_i(f)$ is a power of a linear polynomial. One approach to proving
the conjecture is to first prove it for polynomials of some small degree $d$,
compile a list of bad primes for that degree (namely, those primes $p$ for
which the conjecture fails in degree $d$ and characteristic $p$) and then
deduce the conjecture for all degrees of the form $dp^\ell$,
$\ell\in\mathbb{N}$, where $p$ is a good prime for $d$. In this paper we
calculate certain distinguished monomials appearing in the resultant
$R(f,H_i(f))$ and obtain a (non-exhaustive) list of bad primes for every degree
$d\in\mathbb{N}\setminus\{0\}$.
