Computing Characteristic Polynomials of p-Curvatures in Average Polynomial Time

TL;DR

The paper introduces a fast algorithm for computing characteristic polynomials of p-curvatures of linear differential operators with polynomial coefficients, achieving quasi-linear complexity in the prime bound N, with practical implementations and applications.

Contribution

It presents a novel algorithm that computes all characteristic polynomials of p-curvatures for primes less than N efficiently, improving over previous methods.

Findings

Algorithm operates in asymptotically quasi-linear bit complexity in N.

Implementation results demonstrate rapid performance gains.

Applicable to various problems involving differential operators and p-curvatures.

Abstract

We design a fast algorithm that computes, for a given linear differential operator with coefficients in $Z[x ]$, all the characteristic polynomials of its p-curvatures, for all primes $p < N$ , in asymptotically quasi-linear bit complexity in N. We discuss implementations and applications of our algorithm. We shall see in particular that the good performances of our algorithm are quickly visible.