A deterministic algorithm for finding $r$-power divisors

Building on work of Boneh, Durfee and Howgrave-Graham, we present a deterministic algorithm that provably finds all integers $p$ such that $p^r \mathrel| N$ in time $O(N^{1/4r+\epsilon})$ for any $\epsilon > 0$. For example, the algorithm can be used to test squarefreeness of $N$ in time $O(N^{1/8+\epsilon})$; previously, the best rigorous bound for this problem was $O(N^{1/6+\epsilon})$, achieved via the Pollard--Strassen method.