@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.},
}