Fast algorithms for differential equations in positive characteristic

Abstract

We address complexity issues for linear differential equations in characteristic p >;0: resolution and computation of the p-curvature. For these tasks, our main focus is on algorithms whose complexity behaves well with respect to p. We prove bounds linear in p on the degree of polynomial solutions and propose algorithms for testing the existence of polynomial solutions in sublinear time O(p1/2), and for determining a whole basis of the solution space in quasi-linear time O(p); the O notation indicates that we hide logarithmic factors. We show that for equations of arbitrary order, the p-curvature can be computed in subquadratic time O(p1.79), and that this can be improved to O(log(p)) for first order equations and to O(p) for classes of second order equations.