The dynatomic curves for unimodel polynomials are smooth and irreducible

TL;DR

This paper proves that dynatomic curves for unimodal polynomials are both smooth and irreducible, extending previous results from quadratic to higher degrees using elementary methods and properties of kneading sequences.

Contribution

It establishes the smoothness and irreducibility of dynatomic curves for all degrees d ≥ 2, generalizing prior quadratic cases with simplified proofs.

Findings

Dynatomic curves are smooth for all degrees d ≥ 2.

Dynatomic curves are irreducible for all degrees d ≥ 2.

Elementary methods suffice to prove these properties, avoiding complex internal address techniques.

Abstract

We prove here the smoothness and the irreducibility of the periodic dynatomic curves $ (c,z)\in \C^2$ such that $z$ is $n$-periodic for $z^d+c$, where $d\geq2$. We use the method provided by Xavier Buff and Tan Lei in \cite{BT} where they prove the conclusion for $d=2$. The proof for smoothness is based on elementary calculations on the pushforwards of specific quadratic differentials, following Thurston and Epstein, while the proof for irreducibility is a simplified version of Lau-Schleicher's proof by using elementary arithmetic properties of kneading sequence instead of internal addresses.