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


Authors: Wouter Castryck, Robert Laterveer and Myriam Ounaïes
Journal: Math. Comp. 83 (2014), 3017-3037
MSC (2010): Primary 12D10; Secondary 13Y05, 14Q99
DOI: https://doi.org/10.1090/S0025-5718-2014-02809-3
Published electronically: February 25, 2014
MathSciNet review: 3246822
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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$-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 $ 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).


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Additional Information

Wouter Castryck
Affiliation: Departement Wiskunde, KU Leuven, Celestijnenlaan 200B, 3001 Leuven (Heverlee), Belgium
Email: wouter.castryck@wis.kuleuven.be

Robert Laterveer
Affiliation: Institut de Recherche Mathématique Avancée, Université de Strasbourg, 7 Rue René Descartes, 67084 Strasbourg CEDEX, France
Email: robert.laterveer@math.unistra.fr

Myriam Ounaïes
Affiliation: Institut de Recherche Mathématique Avancée, Université de Strasbourg, 7 Rue René Descartes, 67084 Strasbourg CEDEX, France
Email: myriam.ounaies@unistra.fr

DOI: https://doi.org/10.1090/S0025-5718-2014-02809-3
Received by editor(s): August 27, 2012
Received by editor(s) in revised form: February 7, 2013, and February 12, 2013
Published electronically: February 25, 2014
Article copyright: © Copyright 2014 American Mathematical Society