Families of polynomials of every degree with no rational preperiodic points

Sadek, Mohammad

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

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 $[K:\mathbb{Q}]$. Cependant, les seuls exemples de familles paramétriques de polynômes sans points prépériodiques supposent $2|d$ ou $3|d$ et $K=\mathbb{Q}$. 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 $t\in K$.

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 $[K:\mathbb{Q}]$. 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 $K=\mathbb{Q}$. 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 $t\in K$.

Classification : 37P05, 37P15

[1] Cohen, S. D. The distribution of Galois groups and Hilbert’s irreducibility theorem, Proc. Lond. Math. Soc., Volume 43 (1981), pp. 227-250 | DOI | MR | Zbl

[2] Flynn, Eugene V.; Poonen, Bjorn; Schaefer, Edward F. Cycles of quadratic polynomials and rational points on a genus-2 curve, Duke Math. J., Volume 90 (1997) no. 3, pp. 435-463 | DOI | MR | Zbl

[3] Ingram, Patrick Canonical heights and preperiodic points for certain weighted homogeneous families of polynomials, Int. Math. Res. Not., Volume 2019 (2019) no. 15, pp. 4859-4879 | DOI | MR

[4] Legrand, François Twists of superelliptic curves without rational points, Int. Math. Res. Not., Volume 2018 (2018) no. 4, pp. 1153-1176 | MR | Zbl

[5] Morton, Patrick Arithmetic properties of periodic points of quadratic maps. II, Acta Arith., Volume 87 (1998) no. 2, pp. 89-102 | DOI | MR | Zbl

[6] Morton, Patrick; Silverman, Joseph H. Rational periodic points of rational functions, Int. Math. Res. Not., Volume 1994 (1994) no. 2, pp. 97-110 | DOI | MR | Zbl

[7] Narkiewicz, Władysław Cycle-lengths of a class of monic binomials, Funct. Approximatio, Comment. Math., Volume 42 (2010) no. 2, pp. 163-168 | DOI | MR | Zbl

[8] Narkiewicz, Władysław On a class of monic binomials, Proc. Steklov Inst. Math., Volume 280 (2013) no. 2, pp. 64-69 | MR | Zbl

[9] Poonen, Bjorn The classification of rational preperiodic points of quadratic polynomials over $\mathbb{Q}$: a refined conjecture, Math. Z., Volume 228 (1998) no. 1, pp. 11-29 | DOI | MR | Zbl

[10] Stoll, Michael ational 6-cycles under iteration of quadratic polynomials, LMS J. Comput. Math., Volume 11 (2008), pp. 367-380 | DOI | MR | Zbl