Popov Form Computation for Matrices of Ore Polynomials

Let F[∂;σ ,δ] be a ring of Ore polynomials over a field. We give a new deterministic algorithm for computing the Popov form P of a non-singular matrix A ∈ F[∂;σ ,δ]n×n . Our main focus is to ensure controlled growth in the size of coefficients from F in the case F = k(z), and even k = Q. Our algorithms are based on constructing from A a linear system over F and performing a structured fraction-free Gaussian elimination. The algorithm is output sensitive, with a cost that depends on the orthogonality defect of the input matrix: the sum of the row degrees in Aminus the sum of the row degrees in P . The resulting bit-complexity for the differential and shift polynomial case over Q (z) improves upon the previous best.