Computation of the Similarity Class of the $p$-Curvature

Abstract : The $p$-curvature of a system of linear differential equations in positive characteristic $p$ is a matrix that measures how far the system is from having a basis of polynomial solutions. We show that the similarity class of the $p$-curvature can be determined without computing the $p$-curvature itself. More precisely, we design an algorithm that computes the invariant factors of the $p$-curvature in time quasi-linear in $\sqrt p$. This is much less than the size of the $p$-curvature, which is generally linear in $p$. The new algorithm allows to answer a question originating from the study of the Ising model in statistical physics.
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Submitted on : Tuesday, May 24, 2016 - 5:39:06 PM
Last modification on : Thursday, November 15, 2018 - 11:56:37 AM

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Alin Bostan, Xavier Caruso, Éric Schost. Computation of the Similarity Class of the $p$-Curvature. ISSAC 2016, Jul 2016, Waterloo, ON, Canada. ACM Press, pp.111-118, 〈http://www.issac-conference.org/2016/〉. 〈10.1145/2930889.2930897〉. 〈hal-01321043〉

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