Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algebraic solutions of linear differential equations: an arithmetic approach

Alin Bostan, Xavier Caruso, Julien Roques

TL;DR

This survey argues that algebraicity of solutions to linear differential equations over $\mathbb{Q}(x)$ can be detected via a local-global $p$-curvature paradigm inspired by Grothendieck. It surveys broad instances where algebraic solutions arise (special functions, diagonals, combinatorics, number theory) and articulates Grothendieck’s conjecture that a full basis of algebraic solutions is equivalent to vanishing $p$-curvature for almost all primes. The text details order-1 cases, extends to higher orders through Cartier’s lemma, and reviews Beukers–Heckman results for hypergeometric equations, Katz–Andre progress, and algorithmic methods for computing $p$-curvature. It also connects algebraicity to integrality and p-adic analytic behavior, highlighting both theoretical depth and computational approaches in understanding periods, diagonals, and special function equations. The framework provides a powerful arithmetic lens for recognizing when D-finite functions are algebraic and for guiding effective computations in differential Galois theory.

Abstract

Given a linear differential equation with coefficients in $\mathbb{Q}(x)$, an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic. After presenting motivating examples coming from various branches of mathematics, we advertise in an elementary way a beautiful local-global arithmetic approach to these questions, initiated by Grothendieck in the late sixties. This approach has deep ramifications and leads to the still unsolved Grothendieck-Katz $p$-curvature conjecture.

Algebraic solutions of linear differential equations: an arithmetic approach

TL;DR

This survey argues that algebraicity of solutions to linear differential equations over can be detected via a local-global -curvature paradigm inspired by Grothendieck. It surveys broad instances where algebraic solutions arise (special functions, diagonals, combinatorics, number theory) and articulates Grothendieck’s conjecture that a full basis of algebraic solutions is equivalent to vanishing -curvature for almost all primes. The text details order-1 cases, extends to higher orders through Cartier’s lemma, and reviews Beukers–Heckman results for hypergeometric equations, Katz–Andre progress, and algorithmic methods for computing -curvature. It also connects algebraicity to integrality and p-adic analytic behavior, highlighting both theoretical depth and computational approaches in understanding periods, diagonals, and special function equations. The framework provides a powerful arithmetic lens for recognizing when D-finite functions are algebraic and for guiding effective computations in differential Galois theory.

Abstract

Given a linear differential equation with coefficients in , an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic. After presenting motivating examples coming from various branches of mathematics, we advertise in an elementary way a beautiful local-global arithmetic approach to these questions, initiated by Grothendieck in the late sixties. This approach has deep ramifications and leads to the still unsolved Grothendieck-Katz -curvature conjecture.
Paper Structure (30 sections, 28 theorems, 158 equations, 2 figures)

This paper contains 30 sections, 28 theorems, 158 equations, 2 figures.

Table of Contents

  1. Context, motivation and basic examples
  2. Several natural differential equations have algebraic solutions
  3. Examples from Special functions: Hypergeometric functions
  4. Elliptic integrals: Euler's differential equation
  5. Elliptic integrals: Legendre's differential equation
  6. Gauss' hypergeometric functions
  7. Examples from Algebra: Diagonals
  8. Examples from Combinatorics: Walks in the Quarter Plane
  9. Trident walks
  10. Gessel walks
  11. Examples from Number Theory
  12. Grothendieck's conjecture
  13. The case of equations of order $1$
  14. Rational and algebraic solutions in characteristic $0$: a criterion
  15. Algebraic solutions: from characteristic $0$ to characteristic $p$
  16. ...and 15 more sections

Key Result

Proposition 1.3

Let $f(x)=\sum_{n \geq 0} a_n x^n \in \mathbb{Q}[[x]] \setminus \mathbb{Q}[x]$ be an algebraic non-polynomial function. Then $f(x)$ has a finite number of singularities, a finite nonzero radius of convergence, and its coefficient sequence $(a_n)_{n\geq 0}$ is such that where $m\in \mathbb Z_{\geq 0}$, $r\in\mathbb{Q}\setminus\mathbb Z_{<0}$, $q<r$, $\beta \in \overline{\mathbb{Q}}_{>0}$, and $C_i

Figures (2)

  • Figure 1: A Dyck path
  • Figure 2: Pairs $(u,v)$ for which the sequence $(u)_n \cdot (v)_n \cdot c_n$ is (conjecturally) almost integral. In all cases, the generating functions are (conjecturally) algebraic. Algebraicity degrees are "guessed" by numerical monodromy computations.

Theorems & Definitions (72)

  • Example 1.1: Catalan numbers
  • Example 1.2
  • Proposition 1.3: Theorems A and D in Flajolet87
  • Corollary 1.4: "Flajolet's criterion"
  • Proposition 1.6: "Eisenstein's criterion" (1852)
  • Proposition 1.7
  • Proposition 1.8
  • proof
  • Example 1.9
  • Example 1.10
  • ...and 62 more