A Polynomial-Time Algorithm for the Jacobson Form of a Matrix of Ore Polynomials

Abstract

We present a new algorithm to compute the Jacobson form of a matrix A of polynomials over the Ore domain F(z)[x;σ,δ]^n×n, for a field F. The algorithm produces unimodular U, V and the diagonal Jacobson form J such that UAV = J. It requires time polynomial in deg _x (A), deg _z (A) and n. We also present tight bounds on the degrees of entries in U, V and J. The algorithm is probabilistic of the Las Vegas type: we assume we are able to generate random elements of F at unit cost, and will always produces correct output within the expected time. The main idea is that a randomized, unimodular, preconditioning of A will have a Hermite form whose diagonal is equal to that of the Jacobson form. From this the reduction to the Jacobson form is easy. Polynomial-time algorithms for the Hermite form have already been established.