Constraints on counterexamples to the Casas-Alvero conjecture and a verification in degree $12$
| Autor: | Robert Laterveer, Wouter Castryck, Myriam Ounaïes |
|---|---|
| Přispěvatelé: | Catholic University of Leuven - Katholieke Universiteit Leuven (KU Leuven), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS) |
| Jazyk: | angličtina |
| Rok vydání: | 2014 |
| Předmět: |
12D10
14XX 14-04 30C15 30E99 Algebra and Number Theory Conjecture Mathematics - Complex Variables Applied Mathematics [MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV] Extension (predicate logic) Open case Combinatorics Computational Mathematics Gröbner basis Mathematics - Algebraic Geometry Casas-Alvero conjecture FOS: Mathematics In degree Complex Variables (math.CV) Algebraic Geometry (math.AG) Mathematics Counterexample |
| Zdroj: | Mathematics of Computation Mathematics of Computation, American Mathematical Society, 2014, 83 (290), pp.3017-3037. ⟨10.1090/S0025-5718-2014-02809-3⟩ |
| ISSN: | 0025-5718 |
| DOI: | 10.1090/S0025-5718-2014-02809-3⟩ |
| Popis: | 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). |
| Databáze: | OpenAIRE |
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