Families of polynomials of every degree with no rational preperiodic points

Mohammad Sadek

doi:10.5802/crmath.173

Dynamical systems

Families of polynomials of every degree with no rational preperiodic points

Mohammad Sadek ¹

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

Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 195-197

Abstract (Original language)
Résumé

Let $K$ be a number field. Given a polynomial $f (x) \in K [x]$ of degree $d \geq 2$ , it is conjectured that the number of preperiodic points of $f$ is bounded by a uniform bound that depends only on $d$ and $[K : ℚ]$ . 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 $d \geq 2$ , we display infinitely many parametric families of polynomials of the form $f_{t} (x) = x^{d} + c (t)$ , $c (t) \in K (t)$ , with no rational preperiodic points for any $t \in K$ .

Soit $K$ un corps de nombres. Étant donné un polynôme $f (x) \in K [x]$ de degré $d \geq 2$ , il est conjecturé que le nombre de points prépériodiques de $f$ est borné par une constante ne dépendant que de $d$ et $[K : ℚ]$ . 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 $d \geq 2$ , nous démontrons qu’il existe une infinité de familles paramétriques de polynômes de la forme $f_{t} (x) = x^{d} + c (t)$ , $c (t) \in K (t)$ , sans points prépériodiques rationnels pour tout $t \in K$ .

Article information
Cite this article

Received: 2020-02-07
Revised: 2020-07-06
Accepted: 2020-12-17
Published online: 2021-03-17

DOI: 10.5802/crmath.173

Classification: 37P05, 37P15

Author's affiliations:

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

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

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