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

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.