@inproceedings{BoSc09b,
    author      = {Bostan, Alin and Schost, {\'E}ric},
    title       = {Fast algorithms for differential equations in positive
  characteristic},
    booktitle   = {ISSAC'09},
    year        = {2009},
    editor      = {May, John},    
    pages 	= {47--54},
    doi 	= {10.1145/1576702.1576712},
    arxiv       = {abs/ 0901.3843},
    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 $\tilde{O}(p^{1/2})$, and
 for determining a whole basis of the solution space in quasi-linear time
 $\tilde{O}(p)$; the $\tilde{O}$ notation indicates that we hide logarithmic
 factors. We show that for equations of arbitrary order, the $p$-curvature can
 be computed in subquadratic time $\tilde{O}(p^{1.79})$, and that this can be
 improved to $O(\log(p))$ for first order equations and to $\tilde{O}(p)$ for
 classes of second order equations.},
}