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Let $k$ be a field, $d$ a strictly positive integer and $x$ an independent variable. Let $f \in k[x]$ be a monic polynomial of degree $d$. For $i \in \{1, \dots , d - 1 \}$, let $f^{(i)}$ denote the $i$-th derivative of $f$ (the $i$-th Hasse derivative in case char $k > 0$). Assume that for all $i \in \{1, \dots , d - 1 \}$ the polynomial $f^{(i)}$ has a non-trivial common factor with $f$.

The Casas–Alvero conjecture asserts that, assuming char $k = 0$, there exists $a \in k$ such that $f(x) = (x - a)^d$.

If char$(k) = p > 0$, the conjecture is false in general.

Let us write $\text{CA}_{d,p}$ for the statement ”The Casas-Alvero conjecture holds for polynomials of degree $d$ over fields of characteristic $p$”.

The following equivalences are known for each $d$: 

$\text{CA}_{d,0}$ holds $\iff \text{CA}_{d,p}$ holds for some prime number $p \iff \text{CA}_{d,p}$ holds for all but finitely many primes $p$.

A prime number $p$ is said to be a bad prime for $d$ if $\text{CA}_{d,p}$ is false.

In this talk we will discuss an approach to the conjecture that consists in first proving it for some small degree $d$, compiling lists of bad and good primes for that $d$ and then deducing the conjecture for all the degrees of the form $ dp^l$, where $l$ is a positive integer and $d$ a good prime for $d$. At the end of the talk we will mention a recent result (joint with D. Schaub) that gives a long (but not exhaustive) list of good primes for each $d$.