Connection Matrices in Macaulay2
Paul Görlach, Joris Koefler, Anna-Laura Sattelberger, Mahrud Sayrafi, Hendrik Schroeder, Nicolas Weiss, Francesca Zaffalon
TL;DR
The paper addresses computing connection matrices for holonomic left ideals in the Weyl algebra by leveraging elimination term orders in the rational Weyl algebra and implementing a robust normal-form framework. It introduces algorithms that derive the matrix $A_i$ from normal forms of $\partial_i s_j$ relative to a Gröbner basis, and it formalizes gauge transformations to relate different $\mathbb{C}(x)$-bases. The authors provide a comprehensive Macaulay2 package, ConnectionMatrices, with commands for building elimination-ordered Weyl algebras, computing normal forms, standard monomials, connection matrices, and integrability checks, plus gauge transforms and epsilon-factorization support, demonstrated on representative D-ideals. This work enables systematic construction of Pfaffian systems from $D$-ideals, supports parameter dependencies and dimensional-regularization workflows, and offers public access via the MathRepo implementation, facilitating use in symbolic PDE analysis and related physics applications.
Abstract
In this article, we describe the theoretical foundations of the Macaulay2 package ConnectionMatrices and explain how to use it. For a left ideal in the Weyl algebra that is of finite holonomic rank, we implement the computation of the encoded system of linear PDEs in connection form with respect to an elimination term order that depends on a chosen positive weight vector. We also implement the gauge transformation for carrying out a change of basis over the field of rational functions. We demonstrate all implemented algorithms with examples.