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Degree bounds for linear differential equations and recurrences

Louis Gaillard

TL;DR

This work introduces a unified framework to derive tight degree bounds for a broad class of problems involving D-finite functions and P-recursive sequences, all expressible via pseudo-linear maps. Central to the approach is bounding the McMillan degree of the associated matrix $T$ through a realisation $T=XM^{-1}Y$, which controls the denominators and hence the degree of polynomial relations arising in pseudo-Krylov systems. The authors apply this to a range of closure operations (LCLMs, symmetric products, polynomials), differential equations for algebraic functions, compositions of algebraic and D-finite functions, and Hermite-based creative telescoping, yielding order-degree curves and bounds that improve or recover prior results. The framework provides both theoretical bounds and a practical route for designing algorithms whose performance scales with the size of a compact realisation of $T$, offering potential advances in symbolic computation and automatic proving of differential equations for special functions. Overall, the paper delivers a cohesive, scalable methodology for degree analysis across diverse problems in D-finite theory and recurrence computation.

Abstract

Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing algorithms that compute such representations as a linear relation between the iterates of an elementary operator known as a \emph{pseudo-linear map}. Algorithms of this form have been designed and used for solving various computational problems, in different contexts, including effective closure properties for linear differential or recurrence equations, the computation of a differential equation satisfied by an algebraic function, and many others. We propose a unified approach for establishing precise degree bounds on the solutions of all these problems. This approach relies on a common structure shared by all the specific instances of the class. For each problem, the obtained bound is tight. It either improves or recovers the previous best known bound that was derived by \emph{ad hoc} methods.

Degree bounds for linear differential equations and recurrences

TL;DR

This work introduces a unified framework to derive tight degree bounds for a broad class of problems involving D-finite functions and P-recursive sequences, all expressible via pseudo-linear maps. Central to the approach is bounding the McMillan degree of the associated matrix through a realisation , which controls the denominators and hence the degree of polynomial relations arising in pseudo-Krylov systems. The authors apply this to a range of closure operations (LCLMs, symmetric products, polynomials), differential equations for algebraic functions, compositions of algebraic and D-finite functions, and Hermite-based creative telescoping, yielding order-degree curves and bounds that improve or recover prior results. The framework provides both theoretical bounds and a practical route for designing algorithms whose performance scales with the size of a compact realisation of , offering potential advances in symbolic computation and automatic proving of differential equations for special functions. Overall, the paper delivers a cohesive, scalable methodology for degree analysis across diverse problems in D-finite theory and recurrence computation.

Abstract

Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing algorithms that compute such representations as a linear relation between the iterates of an elementary operator known as a \emph{pseudo-linear map}. Algorithms of this form have been designed and used for solving various computational problems, in different contexts, including effective closure properties for linear differential or recurrence equations, the computation of a differential equation satisfied by an algebraic function, and many others. We propose a unified approach for establishing precise degree bounds on the solutions of all these problems. This approach relies on a common structure shared by all the specific instances of the class. For each problem, the obtained bound is tight. It either improves or recovers the previous best known bound that was derived by \emph{ad hoc} methods.
Paper Structure (37 sections, 26 theorems, 75 equations, 1 table)

This paper contains 37 sections, 26 theorems, 75 equations, 1 table.

Table of Contents

  1. Introduction
  2. Overview of the method
  3. Results in specific instances
  4. Matrices of rational functions
  5. Determinantal denominators and Smith-McMillan normal form
  6. Matrix fraction description
  7. McMillan degree
  8. Proof of Theorem \ref{['thm:McMillandeg_kronecker']}
  9. Degree bounds for pseudo-Krylov systems
  10. Determinantal denominators of pseudo-Krylov matrices
  11. Differential case
  12. Shift case
  13. Bounds on McMillan degree
  14. Differential case
  15. Shift case
  16. ...and 22 more sections

Key Result

Theorem 1

Let $\theta$ be a pseudo-linear map of the shape defined above with a matrix $T \in k(x)^{n \times n}$ ($\theta = p(x)\partial_x + T$ with $p \in k[x]$ of degree $\le 1$, or $\theta = T \cdot \sigma_x$). Let $a \in k[x]^n$ of degree $d_a$ and $\rho = \dim \mathop{\mathrm{span}}\nolimits_{k(x)}(\thet In particular, there exists a solution $\eta \in k[x]^{\rho+1}\setminus \{0\}$ of Problem pb:lin_re

Theorems & Definitions (40)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Theorem 3
  • proof : Proof of Theorem \ref{['thm:degree_bound_pseudo_krylov']}
  • ...and 30 more