FoCM 2014 conference

Workshop B7 - Symbolic Analysis

December 16, 14:30 ~ 14:55 - Room C21

## Fast algorithms for the $p$-curvature of differential operators

### Alin Bostan

### INRIA, France - alin.bostan@inria.fr

The $p$-curvature of a linear differential operator in characteristic $p$ is a matrix that measures to what extent the solution space of the operator has dimension close to its order. We describe a recent algorithm for computing the characteristic polynomial of the $p$-curvature in time ${O\tilde{~}}(p^{0.5})$. The new algorithm 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.

Joint work with Xavier Caruso (Université Rennes 1, France) and Éric Schost (University of Western Ontario, Canada).