Computation of the Similarity Class of the p-Curvature

TL;DR

This paper introduces an efficient algorithm to determine the similarity class of the p-curvature in linear differential systems over positive characteristic fields, significantly reducing computational complexity and enabling applications in statistical physics.

Contribution

The paper presents a novel algorithm that computes the invariant factors of the p-curvature without explicitly calculating it, with a quasi-linear time complexity in p.

Findings

Algorithm computes invariant factors in quasi-linear time in p

Reduces computational complexity compared to direct p-curvature calculation

Enables applications in the study of the Ising model

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.