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Comptes Rendus. Mathématique
Dynamical systems
Families of polynomials of every degree with no rational preperiodic points
Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 195-197.

Let K be a number field. Given a polynomial f(x)K[x] of degree d2, it is conjectured that the number of preperiodic points of f is bounded by a uniform bound that depends only on d and [K:]. However, the only examples of parametric families of polynomials with no preperiodic points are known when d is divisible by either 2 or 3 and K=. In this article, given any integer d2, we display infinitely many parametric families of polynomials of the form f t (x)=x d +c(t), c(t)K(t), with no rational preperiodic points for any tK.

Soit K un corps de nombres. Étant donné un polynôme f(x)K[x] de degré d2, il est conjecturé que le nombre de points prépériodiques de f est borné par une constante ne dépendant que de d et [K:]. Cependant, les seuls exemples de familles paramétriques de polynômes sans points prépériodiques supposent 2|d ou 3|d et K=. Dans cet article, étant donné un entier d2, nous démontrons qu’il existe une infinité de familles paramétriques de polynômes de la forme f t (x)=x d +c(t), c(t)K(t), sans points prépériodiques rationnels pour tout tK.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.173
Classification: 37P05,  37P15
Mohammad Sadek 1

1 Faculty of Engineering and Natural Sciences, Sabancı University, Tuzla, İstanbul, 34956 Turkey
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Mohammad Sadek. Families of polynomials of every degree with no rational preperiodic points. Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 195-197. doi : 10.5802/crmath.173. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.173/

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