Connection Matrices in Macaulay2

Systems of linear PDEs can be encoded as $D$-ideals. When the ideal is holonomic, they can be expressed in terms of first-order matrix differential equations using connection matrices. In some cases, changes of basis (a gauge transform) allow to make the system more manageable, as, for example, with the $\epsilon$-factorized form in physics. To algorithmically compute connection matrices and such gauge transforms, one requires both Gröbner bases and the normal form algorithm in the rational Weyl algebra. We discuss the implementation of those in our new M2-package $\tt ConnectionMatrices$ and showcase its functionalities in examples.

This is joint work with: Paul Görlach, Joris Koefler, Anna-Laura Sattelberger, Mahrud Sayrafi, Hendrik Schroeder, and Francesca Zaffalon.