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Section: New Results

Quasi-optimal computation of the $p$-curvature

The $p$-curvature of a system of linear differential equations in positive characteristic $p$ is a matrix that measures to what extent the system is close to having a fundamental matrix of rational function solutions. This notion, originally introduced in the arithmetic theory of differential equations, has been recently used as an effective tool in computer algebra and in combinatorial applications. We have described in [6] a recent algorithm for computing the $p$-curvature, whose complexity is almost optimal with respect to the size of the output. The new algorithm performs remarkably well in practice. Its design relies on the existence of a well-suited ring, of so-called Hurwitz series, for which an analogue of the Cauchy–Lipschitz Theorem holds, and on a FFT-like method in which the “evaluation points” are Hurwitz series.