A Fast Algorithm for Computing the p-Curvature

TL;DR

This paper introduces a fast, quasi-optimal algorithm for computing the p-curvature of differential systems in positive characteristic, leveraging series rings with divided powers to improve computational efficiency.

Contribution

It presents a novel algorithm with complexity $ ilde{O}(p d r^ ext{omega})$, significantly advancing the computational methods for p-curvature in differential systems.

Findings

Algorithm achieves near-optimal complexity assuming matrix multiplication exponent is 2.

Complexity scales linearly with p and degree d, quadratically with system dimension r.

Provides theoretical foundation using series rings with divided powers.

Abstract

We design an algorithm for computing the $p$-curvature of a differential system in positive characteristic $p$. For a system of dimension $r$ with coefficients of degree at most $d$, its complexity is $\softO (p d r^\omega)$ operations in the ground field (where $\omega$ denotes the exponent of matrix multiplication), whereas the size of the output is about $p d r^2$. Our algorithm is then quasi-optimal assuming that matrix multiplication is (\emph{i.e.} $\omega = 2$). The main theoretical input we are using is the existence of a well-suited ring of series with divided powers for which an analogue of the Cauchy--Lipschitz Theorem holds.