Computing a holonomic submodule of the partial Weyl closure

TL;DR

This paper introduces a new algorithm for computing holonomic submodules of the partial Weyl closure, enhancing symbolic analysis of differential systems with significant speed improvements.

Contribution

It presents a novel algorithm leveraging a non-commutative Rabinowitsch's trick, implemented in Julia, for faster computation of Weyl closures in algebraic analysis.

Findings

Algorithm achieves substantial speedups over existing methods.

Implementation in Julia demonstrates practical efficiency.

Applicable to systems with rational coefficients for symbolic integration.

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.