  In a first (theoretical) part of this paper, we prove a number of constraints
on hypothetical counterexamples to the Casas-Alvero conjecture, building on
ideas of Graf von Bothmer, Labs, Schicho and van de Woestijne that were
recently reinterpreted by Draisma and de Jong in terms of $p$-adic valuations.
In a second (computational) part, we present ideas improving upon Diaz-Toca and
Gonzalez-Vega's Gr\"obner basis approach to the Casas-Alvero conjecture. One
application is an extension of the proof of Graf von Bothmer et al. to the
cases $5p^k$, $6p^k$ and $7p^k$ (that is, for each of these cases, we elaborate
the finite list of primes $p$ for which their proof is not applicable).
Finally, by combining both parts, we settle the Casas-Alvero conjecture in
degree 12 (the smallest open case).
