Section: New Results
Efficient algorithms for linear differential equations in positive characteristic
The -curvature of a linear differential operator in characteristic 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 -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 -curvature in sublinear time . Prior to this work, the fastest algorithms for this task, and even for the subtask of deciding nilpotency of the -curvature, had had merely slightly subquadratic complexity . The new algorithm is also efficient in practice: it allows to test the nilpotency of the -curvature for primes of order , for which the -curvature itself is impossible to compute using current algorithms.