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

Castryck, Wouter; Laterveer, Robert; Ounaïes, Myriam

doi:10.1090/S0025-5718-2014-02809-3

Constraints on counterexamples to the Casas-Alvero conjecture and a verification in degree $12$
HTML articles powered by AMS MathViewer

by Wouter Castryck, Robert Laterveer and Myriam Ounaïes PDF

Math. Comp. 83 (2014), 3017-3037 Request permission

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).

References

Shi Bai, Pierrick Gaudry, Alexander Kruppa, François Morain, Emmanuel Thomé and Paul Zimmerman, CADO-NFS 1.1, available at \verb"http://cado-nfs.gforge.inria.fr/".
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
Eduardo Casas-Alvero, Higher order polar germs, J. Algebra 240 (2001), no. 1, 326–337. MR 1830556, DOI 10.1006/jabr.2000.8727
Mustapha Chellali and Alain Salinier, La conjecture de Casas Alvero pour les degrés $5p^e$, An. Univ. Dunărea de Jos Galaţi Fasc. II Mat. Fiz. Mec. Teor. 4(35) (2012), no. 1-2, 54–62 (French, with French summary). MR 3136558
Rosa de Frutos, Perspectivas aritméticas para la conjetura de Casas-Alvero, Ph.D. thesis, Universidad de Valladolid, (2013).
Gema M. Diaz-Toca and Laureano Gonzalez-Vega, On analyzing a conjecture about univariate polynomials and their roots by using maple, Proceedings of the Maple Conference 2006, Waterloo (Canada), July 23-26, 2006, pp. 81–98 (2006).
Jan Draisma and Johan P. de Jong, On the Casas-Alvero conjecture, Eur. Math. Soc. Newsl. 80 (2011), 29–33. MR 2848893
Hans-Christian Graf von Bothmer, Oliver Labs, Josef Schicho, and Christiaan van de Woestijne, The Casas-Alvero conjecture for infinitely many degrees, J. Algebra 316 (2007), no. 1, 224–230. MR 2354861, DOI 10.1016/j.jalgebra.2007.06.017
János Kollár, Sharp effective Nullstellensatz, J. Amer. Math. Soc. 1 (1988), no. 4, 963–975. MR 944576, DOI 10.1090/S0894-0347-1988-0944576-7
Adrien-Marie Legendre, Théorie des nombres, Firmin Didot Frères, Paris (1830).
Victor V. Prasolov, Polynomials, Algorithms and Computation in Mathematics, vol. 11, Springer-Verlag, Berlin, 2010. Translated from the 2001 Russian second edition by Dimitry Leites; Paperback edition [of MR2082772]. MR 2683151
Paulo Ribenboim, The theory of classical valuations, Springer Monographs in Mathematics, Springer-Verlag, New York, 1999. MR 1677964, DOI 10.1007/978-1-4612-0551-7
Hendrik Verhoek, Some remarks about a polynomial conjecture of Casas-Alvero, Séminaire Bourbakettes, Paris (2009).