Computing the Hermite Form of a Matrix of Ore Polynomials

Abstract: Let R=F[D;sigma,delta] be the ring of Ore polynomials over a field (or skew field) F, where sigma is a automorphism of F and delta is a sigma-derivation. Given a an m by n matrix A over R, we show how to compute the Hermite form H of A and a unimodular matrix U such that UA=H. The algorithm requires a polynomial number of operations in F in terms of both the dimensions m and n, and the degree of the entries in A. When F=k(z) for some field k, it also requires time polynomial in the degree in z, and if k is the rational numbers Q, it requires time polynomial in the bit length of the coefficients as well. Explicit analyses are provided for the complexity, in particular for the important cases of differential and shift polynomials over Q(z). To accomplish our algorithm, we apply the Dieudonne determinant and quasideterminant theory for Ore polynomial rings to get explicit bounds on the degrees and sizes of entries in H and U.