Mathematics of Computation vol:83 issue:290 pages:3017-3037
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 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 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).