In the famous paper(link) of J.Pila he gave an Algorithm on finding $r^{th}$ root of unity in ${\mathbb{F}_p}^{*}$.
The time complexity of that algorithm is mentioned as $O(\log ^{\Delta}{p})$, where $\Delta$ is dependent only on embedding space of A, the number of equations defining A, addition law and their degrees. Here A refers to an abelian variety over $\mathbb{F}_q$.
I want to find the dependence of $\Delta$ on $r$, more precisely is it exponential or double exponential (As he claims result for only constant $r's$).