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An Effective Version of the $p$-Curvature Conjecture for Order One Differential Equations

Florian Fürnsinn, Lucas Pannier

TL;DR

The paper delivers an explicit, computable version of Kronecker's theorem for first-order differential equations by intertwining Hermite–Padé approximants with Honda’s $p$-curvature results. It proves that an effective finite set of primes suffices to certify algebraicity of solutions of $y'(x)=u(x)y(x)$ with rational coefficients, and provides a concrete algorithm to decide algebraicity using $p$-curvatures, with a detailed complexity analysis. A SageMath implementation demonstrates the approach, discusses practical performance relative to alternative methods, and outlines potential extensions to more general coefficient types. This work offers a concrete, arithmetic pathway to the Grothendieck $p$-curvature conjecture in the order-one setting and yields a practical tool for certifying algebraicity of D-finite functions.

Abstract

We develop an effective version of Kronecker's Theorem on the splitting of polynomials, based on asymptotic arguments proposed by the Chudnovsky brothers, coming from Hermite-Padé approximation. In conjunction with Honda's proof of the $p$-curvature conjecture for order one equations with polynomial coefficients we use this to deduce an effective version of the Grothendieck $p$-curvature conjecture for order one equations. More precisely, we bound the number of primes for which the $p$-curvature of a given differential equation has to vanish in terms of the height and the degree of the coefficients, in order to conclude it has a non-zero algebraic solution. Using this approach, we describe an algorithm that decides algebraicity of solutions of differential equation of order one using $p$-curvatures, and report on an implementation in SageMath.

An Effective Version of the $p$-Curvature Conjecture for Order One Differential Equations

TL;DR

The paper delivers an explicit, computable version of Kronecker's theorem for first-order differential equations by intertwining Hermite–Padé approximants with Honda’s -curvature results. It proves that an effective finite set of primes suffices to certify algebraicity of solutions of with rational coefficients, and provides a concrete algorithm to decide algebraicity using -curvatures, with a detailed complexity analysis. A SageMath implementation demonstrates the approach, discusses practical performance relative to alternative methods, and outlines potential extensions to more general coefficient types. This work offers a concrete, arithmetic pathway to the Grothendieck -curvature conjecture in the order-one setting and yields a practical tool for certifying algebraicity of D-finite functions.

Abstract

We develop an effective version of Kronecker's Theorem on the splitting of polynomials, based on asymptotic arguments proposed by the Chudnovsky brothers, coming from Hermite-Padé approximation. In conjunction with Honda's proof of the -curvature conjecture for order one equations with polynomial coefficients we use this to deduce an effective version of the Grothendieck -curvature conjecture for order one equations. More precisely, we bound the number of primes for which the -curvature of a given differential equation has to vanish in terms of the height and the degree of the coefficients, in order to conclude it has a non-zero algebraic solution. Using this approach, we describe an algorithm that decides algebraicity of solutions of differential equation of order one using -curvatures, and report on an implementation in SageMath.

Paper Structure

This paper contains 30 sections, 27 theorems, 40 equations, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Structure of the Paper.
  3. Acknowledgments.
  4. Reformulations of the Problem
  5. Comparison to other Methods
  6. Finding Rational Roots
  7. The Least Prime That Does Not Split in a Number Field
  8. Indicial Equations
  9. An Effective Version of Kronecker's Theorem
  10. Hermite-Padé Approximants to Binomial Powers and Estimates on Their Remainder
  11. Bounding the Denominator
  12. Bounding the Norm
  13. A Bound on the Number of Primes
  14. Choice of the Number of Functions to Approximate
  15. Choice of the Type in the Approximation
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $R(x)\in \mathbb{Q}[x]$ be an irreducible polynomial. If for almost all prime numbers $p$, the reduction of $R(x)$ modulo $p$ has a root in $\mathbb{F}_p$, then $R(x)$ has a root in $\mathbb{Q}$, hence $R(x)$ is linear.

Theorems & Definitions (53)

  • Theorem 1.1: Kronecker
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: Rothstein, Trager
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Example 2.5
  • proof : Proof of Theorem \ref{['thm:intro2']}
  • Proposition 2.6
  • ...and 43 more