Casas-Alvero猜想反例的约束条件及12次验证
Constraints on counterexamples to the Casas-Alvero conjecture, and a verification in degree 12
摘要：
在本文的第一（理论）部分，我们证明了一些约束条件的假设反例的Casas-Alvero猜想，建立在格拉夫冯Bothmer，实验室，Schicho和范德Woestijne的想法，最近重新解释的Draisma和德容在$p$ -进估值。在第二（计算）部分，我们提出了改进的想法后，迪亚兹-Toca和冈萨雷斯-维加的Gröbner基础上的Casas-Alvero猜想的方法。一个应用是Graf von Bothmer等人的证明扩展到$5p^k$、$6p^k$和$7p^k$的情形（也就是说，对于这些情形中的每一个，我们详细说明了素数$p$的有限列表，对于这些素数，我们的证明是不适用的）。最后，通过结合这两个部分，我们解决了Casas-Alvero猜想在12度（最小的开放情况）。
英文摘要：
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 $5p^k$, $6p^k$ 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).
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