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 afieldF. The algorithm produces unimodular U, V and the diagonal Jacobson form J such that UAV = J. It requires time polynomial in degx(A), degz (A) andn. We also present tight bounds on the degrees of entries in U, V and J. The algorithm is probabilistic of the Las Vegas type:weassumeweareabletogeneraterandomelementsofF 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.