A fast algorithm for computing the characteristic polynomial of the p-curvature

We discuss theoretical and algorithmic questions related to the $p$-curvature of differential operators in characteristic $p$. Given such an operator $L$, and denoting by $\mathrm{X}(L)$ the characteristic polynomial of its $p$-curvature, we first prove a new, alternative, description of $\mathrm{X}(L)$. This description turns out to be particularly well suited to the fast computation of $\mathrm{X}(L)$ when $p$ is large: based on it, we design a new algorithm for computing $\mathrm{X}(L)$, whose cost with respect to $p$ is $\text{softO}(p^{0.5})$ operations in the ground field. This is remarkable since, prior to this work, the fastest algorithms for this task, and even for the subtask of deciding nilpotency of the $p$-curvature, had merely slightly subquadratic complexity $\text{softO}(p^{1.79})$.