Computing Characteristic Polynomials of p-Curvatures in Average Polynomial Time

We design a fast algorithm that computes, for a given linear differential operator with coefficients in 918;[x], all the characteristic polynomials of its p-curvatures, for all primes p < N, in asymptoti- cally quasi-linear bit complexity in N. We discuss implementations and applications of our algorithm. We shall see in particular that the good performances of our algorithm are quickly visible.