Converting between the Popov and the Hermite form of matrices of differential operators using an FGLM-like algorithm

We consider matrices over a ring K [∂; σ , θ] of Ore polynomials over a skew field K . Since the Popov and Hermite normal forms are both Gröbner bases (for term over position and position over term ordering resp.), the classical FGLM-algorithm provides a method of converting one into the other. In this report we investigate the exact formulation of the FGLM algorithm for not necessarily “zero-dimensional” modules and give an illustrating implementation in Maple. In an additional section, we will introduce a second notion of Gröbner bases roughly following [Pau07]. We will show that these vectorial Gröbner bases correspond to row-reduced matrices.