872 research outputs found
A Fast Algorithm for Computing the p-Curvature
We design an algorithm for computing the -curvature of a differential
system in positive characteristic . For a system of dimension with
coefficients of degree at most , its complexity is \softO (p d r^\omega)
operations in the ground field (where denotes the exponent of matrix
multiplication), whereas the size of the output is about . Our
algorithm is then quasi-optimal assuming that matrix multiplication is
(\emph{i.e.} ). 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.Comment: ISSAC 2015, Jul 2015, Bath, United Kingdo
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