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arxiv: 2602.06209 · v2 · submitted 2026-02-05 · 💻 cs.SC

Computing a holonomic submodule of the partial Weyl closure

Hadrien Brochet This is my paper

Pith reviewed 2026-05-16 06:34 UTC · model grok-4.3

classification 💻 cs.SC
keywords Weyl closureholonomic modulesdifferential operatorsRabinowitsch trickpartial closureoperator modulessymbolic computationalgebraic analysis
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The pith

A new algorithm computes a holonomic submodule of the partial Weyl closure using a non-commutative analogue of Rabinowitsch's trick.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces an algorithm that produces a holonomic submodule inside the partial Weyl closure of a finite-rank module over differential operators. The partial closure is taken only with respect to a chosen subset of the variables. This converts a system given by operators with rational coefficients into an equivalent system whose coefficients are polynomials. The construction rests on a non-commutative version of Rabinowitsch's trick that reduces the problem to locating a suitable submodule. The resulting submodule encodes additional information on the singularities of the original system.

Core claim

There exists an algorithm that, given a finite-rank module, returns a holonomic submodule of its partial Weyl closure taken with respect to any chosen subset of the variables; the algorithm proceeds by applying a non-commutative analogue of Rabinowitsch's trick to locate the submodule.

What carries the argument

The non-commutative analogue of Rabinowitsch's trick, which reduces the search for a holonomic submodule inside the partial Weyl closure to an auxiliary computation over an extended ring.

If this is right

  • The computed submodule has polynomial coefficients and is contained in the partial Weyl closure.
  • The submodule remains holonomic when the closure is performed only on a proper subset of the variables.
  • The procedure supplies a concrete algebraic object that can serve as input to downstream algorithms for symbolic integration.
  • The method works for any finite-rank input module that satisfies the standing hypotheses of the closure operation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction might be used to extract annihilators that appear in creative-telescoping problems restricted to certain variables.
  • Running the procedure on modules of increasing rank would give practical bounds on the degree growth of the generators.
  • The construction suggests a possible route to an algorithm for the full (non-partial) Weyl closure by iterating over all variable subsets.
  • The output submodule could be fed directly into existing packages that compute dimensions of solution spaces or de Rham cohomology.

Load-bearing premise

A finite-rank module always possesses a holonomic submodule after the partial closure is taken, and the non-commutative Rabinowitsch trick applies directly without further conditions on the coefficient field or the chosen variable subset.

What would settle it

Apply the algorithm to a concrete finite-rank module, such as the module generated by a single operator with rational coefficients in two variables, compute the candidate submodule, and check whether every generator annihilates a nonzero holonomic function or whether the submodule fails to lie inside the partial Weyl closure.

read the original 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 compute a holonomic submodule of the partial Weyl closure of a finite-rank module, where the closure is taken with respect to a subset of the variables. The method relies on a non-commutative analogue of Rabinowitsch's trick. The algorithm is implemented in the Julia package MultivariateCreativeTelescoping.jl and shows substantial speedups over existing exact Weyl closure algorithms in Singular and Macaulay2.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper introduces a new algorithm to compute a holonomic submodule of the partial Weyl closure (with respect to a subset of variables) of a finite-rank module over the Weyl algebra. The method uses a non-commutative analogue of Rabinowitsch's trick and is implemented in the Julia package MultivariateCreativeTelescoping.jl, with reported substantial speedups over exact Weyl-closure routines in Singular and Macaulay2.

Significance. If the central claim holds, the work supplies a practical and faster tool for a fundamental operation in algebraic analysis. Partial Weyl closures encode refined singularity information and serve as a preparatory step for creative-telescoping and symbolic-integration algorithms; an efficient, implemented method for the partial case would therefore be a useful addition to the D-module computational toolkit.

major comments (1)
  1. [Algorithm description and correctness argument (likely §3)] The algorithm's correctness rests on the non-commutative Rabinowitsch trick producing a holonomic submodule of the partial closure for arbitrary finite-rank modules and arbitrary proper subsets of variables. The manuscript must supply a precise statement of the ring-theoretic hypotheses (Ore conditions, flatness, or coefficient-field restrictions) under which the localization step remains inside the closure and preserves holonomicity; without this justification the central claim cannot be verified.
minor comments (1)
  1. [Implementation and experimental results] The abstract and implementation section should include a brief table or list of the benchmark examples (modules, variable subsets, and timings) so that the reported speedups can be reproduced and compared directly.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting the need for explicit ring-theoretic hypotheses. We address the major comment below and will incorporate the requested clarification in the revised version.

read point-by-point responses

  1. Referee: [Algorithm description and correctness argument (likely §3)] The algorithm's correctness rests on the non-commutative Rabinowitsch trick producing a holonomic submodule of the partial closure for arbitrary finite-rank modules and arbitrary proper subsets of variables. The manuscript must supply a precise statement of the ring-theoretic hypotheses (Ore conditions, flatness, or coefficient-field restrictions) under which the localization step remains inside the closure and preserves holonomicity; without this justification the central claim cannot be verified.

    Authors: We agree that the correctness argument requires an explicit statement of the hypotheses. In the revised manuscript we will insert a new paragraph in §3 stating that the base field K has characteristic zero, that the (partial) Weyl algebra A satisfies the left and right Ore conditions with respect to the multiplicative set generated by the variables outside the chosen subset (allowing localization), and that the input module M is a finite-rank left A-module. Under these conditions the non-commutative Rabinowitsch trick produces a submodule of the partial Weyl closure that remains holonomic. We will also cite the relevant results on Ore localization in Weyl algebras (e.g., the work of McConnell–Robson and subsequent D-module literature) to justify that holonomicity is preserved. revision: yes

Circularity Check

0 steps flagged

No circularity: new algorithm presented as independent construction

full rationale

The paper introduces a new algorithm to compute a holonomic submodule of the partial Weyl closure using a non-commutative analogue of Rabinowitsch's trick, with an implementation in Julia. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain. The central claim is an algorithmic construction whose correctness is asserted independently of the target result, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, axioms, or invented entities are detailed in the provided text.

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Reference graph

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