Factoring Gleason polynomials modulo 2
Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 787-812.

Among the connected components of the interior of the Mandelbrot set are those that are hyperbolic. These components consist of parameters c for which the critical point z 0 =0 of f c :zz 2 +c is attracted to an attracting periodic cycle. Every hyperbolic component contains a unique center; that is, a parameter c for which the critical point z 0 is periodic. For a given n1, the Gleason polynomial for period n is the monic polynomial G n [c] whose roots are exactly the centers of the hyperbolic components of period n. It is unknown if G n factors over . In this article, we factor G n modulo 2. We prove the following remarkable fact: the number of irreducible factors of G n modulo 2 is equal to the number of real roots of G n .

Parmi les composantes connexes de l’intérieur de l’ensemble de Mandelbrot, on trouve celles qui sont hyperboliques. Ces composantes correspondent aux paramètres c pour lesquels le point critique z 0 =0 du polynôme f c :zz 2 +c est attiré par un cycle attractif. Chaque composante hyperbolique contient un unique centre ; c’est le paramètre c pour lequel z 0 est périodique. Étant donné un entier n1, le polynôme de Gleason de période n est le polynôme unitaire G n [c] dont les racines sont précisément les centres des composantes hyperboliques de période n. On ne sait pas si G n se factorise sur . Dans cet article, nous factorisons G n modulo 2. Nous prouvons le fait remarquable suivant : le nombre de facteurs irréductibles de G n modulo 2 est égal au nombre de racines réelles de G n .

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1228
Classification: 37F10, 37P35, 37P05
Keywords: Mandelbrot set, hyperbolic component, Gleason polynomial
Xavier Buff 1; William Floyd 2; Sarah Koch 3; Walter Parry 4

1 Institut de Mathématiques de Toulouse Université Paul Sabatier 118, route de Narbonne 31062 Toulouse Cedex, France
2 Department of Mathematics Virginia Tech Blacksburg, VA 2406, U.S.A.
3 Department of Mathematics University of Michigan Ann Arbor, MI 48109, U.S.A.
4 Department of Mathematics and Statistics Eastern Michigan University Ypsilanti, MI 48197, U.S.A.
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Xavier Buff; William Floyd; Sarah Koch; Walter Parry. Factoring Gleason polynomials modulo 2. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 787-812. doi : 10.5802/jtnb.1228. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1228/

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