Constraints on counterexamples to the Casas-Alvero conjecture and a verification in degree
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 | References | Similar Articles | Additional Information
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 -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
,
and
(that is, for each of these cases, we determine the finite list of primes
to which their proof is not applicable). Finally, by combining both parts, we settle the Casas-Alvero conjecture in degree
(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