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Computing Characteristic Polynomials of p-Curvatures in Average Polynomial Time

Raphaël Pagès

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.

Computing Characteristic Polynomials of p-Curvatures in Average Polynomial Time

Abstract

We design a fast algorithm that computes, for a given linear differential operator with coefficients in , all the characteristic polynomials of its p-curvatures, for all primes , 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.

Paper Structure

This paper contains 16 sections, 16 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Differential operators
  3. Euler and integration operators
  4. Operators and $p$-curvature
  5. Extension to integral coefficients
  6. Main algorithm
  7. Reverse isomorphism, computation modulo $\theta^{d+1}$
  8. Translation before the computation
  9. Computing a matrix factorial modulo $p$ for a large amount of primes $p$
  10. Final algorithm
  11. Implementation and timings
  12. Timings on random operators
  13. Quasilinear as expected.
  14. Comparison with the previous algorithm.
  15. Execution on special operators
  16. ...and 1 more sections

Key Result

Proposition 2.1

The rings $\mathbb{F}_p[x]\langle\partial^{\pm 1}\rangle\subset\mathbb{F}_p(x)\langle\partial^{\pm 1}\rangle$ (resp. $\mathbb{F}_p[\theta]\langle\partial^{\pm1}\rangle \subset\mathbb{F}_p(\theta)\langle\partial^{\pm 1}\rangle$) of Laurent polynomials in the variable $\partial$ are all well defined.

Figures (2)

  • Figure 1: Computation time for random operators of varying orders and degrees
  • Figure 2: Comparison between the iteration of BoCaSc14's algorithm and our algorithm

Theorems & Definitions (36)

  • Proposition 2.1: BoCaSc14
  • proof
  • Theorem 2.2: BoCaSc14
  • proof
  • Remark 2.3: BoCaSc14
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6: BoCaSc14
  • proof
  • Theorem 2.7: BoCaSc14
  • ...and 26 more