  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.
