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

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.

💡 Research Summary

The paper addresses the computational bottleneck associated with Weyl closure, a fundamental operation in algebraic analysis that transforms a system of linear partial differential equations with rational coefficients into an equivalent system with polynomial coefficients. While Weyl closure is essential for symbolic integration and for constructing annihilators of holonomic functions, existing exact algorithms (notably Tsai’s method) rely on a full localization step and the computation of a b‑function, both of which become prohibitively expensive in multivariate or parametric settings.

The authors propose a novel approximation algorithm for the partial Weyl closure, i.e., the closure taken only with respect to a chosen subset of the variables (typically the integration variables). The core ideas are:

1. Non‑commutative Rabinowitsch trick – Introduce a new auxiliary variable T interpreted as 1/p for a polynomial p that vanishes on the singular locus of the input module S. The relation p·T − 1 = 0 is added to the system. Because the commutation rules between T and the differential operators are awkward, Gröbner bases are computed in the infinite‑rank module W