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 √ 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.