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Section: New Results

Efficient algorithms for linear differential equations in positive characteristic

The $p$-curvature of a linear differential operator in characteristic $p$ is a matrix that measures to what extent the space of polynomial solutions of the operator has dimension close to its order. This makes the $p$-curvature a useful tool in concrete applications, like in combinatorics and statistical physics, where it serves for instance as an a posteriori certification filter for differential operators obtained by guessing techniques. In [9] , we designed a new algorithm for computing the characteristic polynomial of the $p$-curvature in sublinear time $Õ\left(p^{0.5}\right)$. Prior to this work, the fastest algorithms for this task, and even for the subtask of deciding nilpotency of the $p$-curvature, had had merely slightly subquadratic complexity $Õ\left(p^{1.79}\right)$. The new algorithm is also efficient in practice: it allows to test the nilpotency of the $p$-curvature for primes $p$ of order ${10}^6$, for which the $p$-curvature itself is impossible to compute using current algorithms.