The dynatomic periodic curves for polynomial z→z~d+c are smooth and irreducible

We prove here the smoothness and the irreducibility of the periodic dynatomic curves(c,z)∈C2such that z is n-periodic for zd+c,where d 2.We use the method provided by Buff and Lei where they proved 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.