Fast algorithms for differential equations in positive characteristic

TL;DR

This paper develops efficient algorithms for solving linear differential equations over fields of positive characteristic, achieving near-linear and subquadratic complexities for key computational tasks related to the $p$-curvature and polynomial solutions.

Contribution

It introduces algorithms with complexity bounds linear or sublinear in $p$ for computing solutions and $p$-curvature, improving efficiency in positive characteristic differential equations.

Findings

Polynomial solutions can be tested in $ ilde{O}(p^{1/2})$ time.

Solution space basis can be computed in $ ilde{O}(p)$ time.

$p$-curvature can be computed in $ ilde{O}(p^{1.79})$ time, with faster methods for specific cases.

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.