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arXiv · 1506.05645

A Fast Algorithm for Computing the p-Curvature

Alin Bostan·Xavier Caruso·Éric Schost
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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.

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Alin Bostan, Xavier Caruso, Éric Schost. 2015-06-18. A Fast Algorithm for Computing the p-Curvature. https://doi.org/10.1145/2755996.2756674

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