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

TL;DR

This paper combines theoretical constraints and computational methods to analyze the Casas-Alvero conjecture, ultimately proving it for degree 12, the smallest unresolved case, by extending previous results and refining algorithms.

Contribution

It introduces new theoretical constraints on counterexamples and improves computational approaches, leading to the proof of the conjecture in degree 12.

Findings

Constraints on counterexamples to the conjecture.

Extension of proofs to specific prime power cases.

Verification of the conjecture in degree 12.

Abstract

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).