On the set of bad primes in the study of the Casas–Alvero conjecture

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 \(d\in \mathbb {N}\setminus \{0\}\).