Constraints on counterexamples to the Casas-Alvero conjecture and a verification in degree $12$

In the 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 the second (computational) part, we present ideas improving upon Diaz-Toca and Gonzalez-Vega's Gröbner 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 determine the finite list of primes $p$ to which their proof is not applicable). Finally, by combining both parts, we settle the Casas-Alvero conjecture in degree $12$ (the smallest open case).