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

Abstract

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 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 5 p k 5p^k , 6 p k 6p^k and 7 p k 7p^k (that is, for each of these cases, we determine the finite list of primes p p to which their proof is not applicable). Finally, by combining both parts, we settle the Casas-Alvero conjecture in degree 12 12 (the smallest open case).