Preperiodic dynatomic curves for $z\mapsto z^d+c$

Yan Gao¹

¹ Mathematical School of Sichuan University ChengDu, 610065, P.R. China

Fundamenta Mathematicae, Tome 233 (2016) no. 1, pp. 37-69

Résumé

The preperiodic dynatomic curve $\mathcal {X}_{n,p}$ is the closure in $\mathbb C^2$ of the set of $(c,z)$ such that $z$ is a preperiodic point of the polynomial $z \mapsto z^d+c$ with preperiod $n$ and period $p$ ($n,p\geq 1$). We prove that each $\mathcal {X}_{n,p}$ has exactly $d-1$ irreducible components, which are all smooth and have pairwise transverse intersections at the singular points of $\mathcal {X}_{n,p}$. We also compute the genus of each component and the Galois group of the defining polynomial of $\mathcal {X}_{n,p}$.

Détail

DOI : 10.4064/fm91-12-2015

Keywords: preperiodic dynatomic curve mathcal closure mathbb set preperiodic point polynomial mapsto preperiod period geq prove each mathcal has exactly d irreducible components which smooth have pairwise transverse intersections singular points nbsp mathcal compute genus each component galois group defining polynomial mathcal