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Computing an approximation of the partial Weyl closure of a holonomic module

Hadrien Brochet

TL;DR

This work extends the Weyl-closure framework to partial closures with respect to a subset of variables by working in mixed Weyl algebras $W_{ t,oldsymbol{x}}( t)$ and proving that the partial closure can be obtained up to holonomicity via inverting a polynomial that vanishes on the singular locus. It introduces a holonomic-approximation algorithm based on a noncommutative Rabinowitsch trick, performing iterative saturations to control growth and terminate when the quotient becomes holonomic. The method is implemented in Julia (MultivariateCreativeTelescoping.jl) and demonstrates substantial speedups over exact Weyl-closure algorithms in Singular and Macaulay2, albeit with the trade-off that the result is an approximation rather than full closure. The approach improves practicality for symbolic integration tasks involving parametric integrals and holonomic annihilators, enabling more scalable computations in the D-module framework. Overall, the paper contributes a principled, computationally efficient pathway to obtain holonomic partial Weyl closures, with clear implications for automated integral computations and symbolic analysis.

Abstract

The Weyl closure is a basic operation in algebraic analysis: it converts a system of differential operators with rational coefficients into an equivalent system with polynomial coefficients. In addition to encoding finer information on the singularities of the system, it serves as a preparatory step for many algorithms in symbolic integration. A new algorithm is introduced to approximate the partial Weyl closure of a holonomic module, where the closure is taken with respect to a subset of the variables. The method is based on a non-commutative generalization of Rabinowitsch's trick and yields a holonomic module included in the Weyl closure of the input system. The algorithm is implemented in the Julia package MultivariateCreativeTelescoping.jl and shows substantial speedups over existing exact Weyl closure algorithms in Singular and Macaulay2.

Computing an approximation of the partial Weyl closure of a holonomic module

TL;DR

This work extends the Weyl-closure framework to partial closures with respect to a subset of variables by working in mixed Weyl algebras and proving that the partial closure can be obtained up to holonomicity via inverting a polynomial that vanishes on the singular locus. It introduces a holonomic-approximation algorithm based on a noncommutative Rabinowitsch trick, performing iterative saturations to control growth and terminate when the quotient becomes holonomic. The method is implemented in Julia (MultivariateCreativeTelescoping.jl) and demonstrates substantial speedups over exact Weyl-closure algorithms in Singular and Macaulay2, albeit with the trade-off that the result is an approximation rather than full closure. The approach improves practicality for symbolic integration tasks involving parametric integrals and holonomic annihilators, enabling more scalable computations in the D-module framework. Overall, the paper contributes a principled, computationally efficient pathway to obtain holonomic partial Weyl closures, with clear implications for automated integral computations and symbolic analysis.

Abstract

The Weyl closure is a basic operation in algebraic analysis: it converts a system of differential operators with rational coefficients into an equivalent system with polynomial coefficients. In addition to encoding finer information on the singularities of the system, it serves as a preparatory step for many algorithms in symbolic integration. A new algorithm is introduced to approximate the partial Weyl closure of a holonomic module, where the closure is taken with respect to a subset of the variables. The method is based on a non-commutative generalization of Rabinowitsch's trick and yields a holonomic module included in the Weyl closure of the input system. The algorithm is implemented in the Julia package MultivariateCreativeTelescoping.jl and shows substantial speedups over existing exact Weyl closure algorithms in Singular and Macaulay2.
Paper Structure (19 sections, 10 theorems, 70 equations, 3 tables)

This paper contains 19 sections, 10 theorems, 70 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions
  4. Preliminaries
  5. Weyl algebras and holonomicity
  6. Polynomial and rational Weyl algebra
  7. Holonomic $W_\mathbf{x}$-modules.
  8. Singular Locus
  9. Principal symbol
  10. Initial module and characteristic variety
  11. Singular locus
  12. Holonomic W(t,x)(t)-modules
  13. Partial Weyl closure by saturation
  14. The algorithm
  15. Timings
  16. ...and 4 more sections

Key Result

Theorem 1

Let $S$ be a non-trivial submodule of $W_\mathbf{x}^r$, let $G$ be a Gröbner basis of $S$, and let $e_1,\dots,e_r$ be the canonical basis of $W_\mathbf{x}^r$. The following statements are equivalent:

Theorems & Definitions (20)

  • Theorem 1: Folklore
  • Definition 2
  • Lemma 3
  • Theorem 4
  • proof
  • Theorem 5
  • Lemma 6
  • proof
  • proof : Proof of \ref{['thm:singlocus2']}.
  • Theorem 7
  • ...and 10 more