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

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

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

Abstract
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 $t \in K$ .

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

@article{CRMATH_2021__359_2_195_0,
     author = {Mohammad Sadek},
     title = {Families of polynomials of every degree with no rational preperiodic points},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {195--197},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {2},
     year = {2021},
     doi = {10.5802/crmath.173},
     language = {en},
}

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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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