Fast algorithms for differential equations in positive characteristic

Alin Bostan (1) , Éric Schost (2)

1 ALGORITHMS - Algorithms
2 Department of Computer Science

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.

Mots clés: Algorithms; complexity; differential equations; polynomial solutions; p-curvature

Domaines: Calcul formel [cs.SC]

Alin Bostan, Éric Schost. Fast algorithms for differential equations in positive characteristic. 2009. ⟨inria-00355818⟩