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Conference Papers Year : 2015

A Fast Algorithm for Computing the p-Curvature

Alin Bostan
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Éric Schost
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Abstract

We design an algorithm for computing the $p$-curvature of adifferential system in positive characteristic $p$. For a system ofdimension $r$ with coefficients of degree at most $d$, its complexityis $\tilde O (p d r^\omega)$ operations in the ground field (where$\omega$ denotes the exponent of matrix multiplication), whereas thesize of the output is about $p d r^2$. Our algorithm is thenquasi-optimal assuming that matrix multiplication is (\emph{i.e.}$\omega = 2$). The main theoretical input we are using is theexistence of a well-suited ring of series with divided powers forwhich an analogue of the Cauchy--Lipschitz Theorem holds.
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Dates and versions

hal-01164471 , version 1 (17-06-2015)

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Alin Bostan, Xavier Caruso, Éric Schost. A Fast Algorithm for Computing the p-Curvature. ISSAC 2015, Jul 2015, Bath, United Kingdom. pp.69-76, ⟨10.1145/2755996.2756674⟩. ⟨hal-01164471⟩
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