Algebraicity and integrality of solutions to differential equations (2501.13175v1)

Abstract: We formulate a conjecture classifying algebraic solutions to (possibly non-linear) algebraic differential equations, in terms of the primes appearing in the denominators of the coefficients of their Taylor expansion at a non-singular point. For linear differential equations, this conjecture is a strengthening of the Grothendieck-Katz $p$-curvature conjecture. We prove the conjecture for many differential equations and initial conditions of algebro-geometric interest. For linear differential equations, we prove it for Picard-Fuchs equations at initial conditions corresponding to cycle classes, among other cases. For non-linear differential equations, we prove it for isomonodromy differential equations, such as the Painlev\'e VI equation and Schlesinger system, at initial conditions corresponding to Picard-Fuchs equations. We draw a number of algebro-geometric consequences from the proofs.

• The paper presents a new conjecture classifying algebraic solutions of differential equations based on the integrality of Taylor coefficient denominators.
• It extends the classical Grothendieck-Katz p-curvature conjecture to non-linear cases, including isomonodromy equations like Painlevé VI.
• The research combines algebraic, arithmetic, and geometric methods, offering robust criteria and pathways for future developments in algebraic geometry.

Overview of the paper "Algebraicity and Integrality of Solutions to Differential Equations"

The paper by Yeuk Hay Joshua Lam and Daniel Litt presents a nuanced investigation into the algebraic nature of solutions to algebraic differential equations. It introduces a conjecture that aims to classify algebraic solutions of these equations, specifically in relation to the primes appearing in the denominators of the coefficients of their Taylor expansions at non-singular points. This conjecture provides a new perspective on the Grothendieck-Katz $p$-curvature conjecture by extending its principles to non-linear differential equations.

Main Contributions and Results

The paper proposes a conjecture for algebraic differential equations, suggesting a detailed classification of algebraic solutions based on the arithmetic properties of their coefficients. The authors provide substantial evidence for their conjecture, proving it for a variety of cases, including both linear and non-linear differential equations. Key achievements in the paper include:

1. Extension of Grothendieck-Katz $p$-curvature Conjecture: For linear differential equations, the conjecture strengthens the classical Grothendieck-Katz $p$-curvature conjecture, incorporating new conditions related to the integrality of the solution's Taylor coefficients.
2. Verification for Non-linear Equations: The conjecture is extended to non-linear differential equations, and the authors prove it for a significant class of these equations. This includes isomonodromy equations like the Painlevé VI equation and Schlesinger systems, evaluated at specific initial conditions.
3. Implications for Algebraic Geometry: The theoretical framework provided by the conjecture and its proofs offer insights into algebraic geometry, particularly regarding the behavior of differential equations defined by algebraic data.
4. Foliation and Integrality: The paper conceptualizes a notion of integrality for foliations over algebraic spaces and connects these ideas with the algebraicity of solutions to differential equations.

Technical Details

The authors articulate their conjecture using precise algebraic formulations and demonstrate significant examples where the conjecture holds:

• Conjecture Postulates: The conjecture posits that the algebraic nature of solutions can be determined by analyzing the integrality properties of the Taylor series coefficients of these solutions.
• Algebraic and $\omega(p)$-integral Leaves: A significant portion of the paper involves classifying solutions that are either algebraic, integral, or $\omega(p)$-integral, with classifications based on how primes appear in the denominators of coefficients.
• Hodge Filtration Extension: For isomonodromy systems, the paper discusses the extension of Hodge filtrations, which are connected to the existence of algebraic solutions.
• Proof Techniques: An intricate blend of algebraic, arithmetic, and geometric techniques is used to verify the conjecture for various instances, including the use of detailed computations of $p$-curvature.

Speculations and Future Developments

The implications of these results are profound both for theory and applications, as they offer robust criteria for predicting the behavior of differential equations. Future developments could focus on extending these results to wider classes of differential equations and exploring implications for computational aspects of algebraic geometry and arithmetic differential equations.

Conclusion

This paper presents a significant advancement in the field by offering deeper understanding into the nature of algebraic solutions of differential equations through arithmetic characteristics of their coefficients. The conjectures and proofs laid out propose new pathways for future research, potentially impacting a variety of mathematical domains related to algebra and geometry.