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 $f\left(x\right)\in K\left[x\right]$ of degree $d\ge 2$, it is conjectured that the number of preperiodic points of $f$ is bounded by a uniform bound that depends only on $d$ and $\left[K:ℚ\right]$. 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 $K=ℚ$. In this article, given any integer $d\ge 2$, we display infinitely many parametric families of polynomials of the form ${f}_{t}\left(x\right)={x}^{d}+c\left(t\right)$, $c\left(t\right)\in K\left(t\right)$, with no rational preperiodic points for any $t\in K$.

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

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

1 Faculty of Engineering and Natural Sciences, Sabancı University, Tuzla, İstanbul, 34956 Turkey
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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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