CONVERSION BETWEEN HERMITE AND POPOV NORMAL FORMS USING AN FGLM-LIKE APPROACH

Abstract

We are working with matrices over a ring $K\left[\partial ;\sigma ,\vartheta \right]$ of Ore polynomials over a skew field $K$. Extending a result of [18] for usual polynomials it is shown that in this setting the Hermite and Popov normal forms correspond to Gröbner bases with respect to certain orders. The FGLM algorithm is adapted to this setting and used for converting Popov forms into Hermite forms and vice versa. The approach works for arbitrary, that is, not necessarily square matrices where we establish termination criteria to deal with infinitely dimensional factor spaces.