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

2026-02-05

AI Summary

This work addresses the efficient computation of partial Weyl closures for parametric differential operator systems—a crucial preprocessing step in symbolic integration and related algorithms—by transforming rational coefficients into polynomial ones. We propose a novel algorithm that generalizes the Rabinowitsch trick within a noncommutative algebraic framework to effectively approximate the Weyl closure of holonomic modules with respect to a subset of variables. The resulting module is guaranteed to be contained within the true closure. To our knowledge, this is the first method to achieve an efficient approximation of partial Weyl closures, substantially avoiding the high computational complexity of exact approaches. Implemented in Julia within the MultivariateCreativeTelescoping.jl package, our method demonstrates significant speedups over exact implementations in Singular and Macaulay2 across multiple test cases, while preserving correctness through guaranteed containment.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Security and Privacy: Applications of cryptography

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.

Problem

• Weyl closure
• holonomic module
• partial closure
• differential operators
• symbolic integration

Innovation

• partial Weyl closure
• holonomic module
• non-commutative Rabinowitsch trick
• symbolic integration
• creative telescoping