Computing Popov Form of Ore Polynomial Matrices

We show that the computation of the Popov form of Ore polynomial matrices can be formulated as a problem of computing the left nullspace of such matrices. While this technique is already known for polynomial matrices, the extension to Ore polynomial matrices is not immediate because multiplication of the matrix entries is not commutative. A number of results for polynomial matrices are extended to Ore polynomial matrices in this paper. This in turn allows nullspace algorithms to be used in Popov form computations when the input matrix has full row rank. In particular, recent fraction-free and modular algorithms for nullspace computation can be used in exact arithmetic setting where coefficient growth is a concern. In the case when the input matrix does not have full row rank, we show that if a degree bound on the minimal unimodular multiplier is known, the Popov form computation can also be reduced to left nullspace computation.