Computation of the Similarity Class of the p-Curvature
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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 √(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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