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

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 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, 6p and 7p (that is, for each of these cases, we elaborate 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).