Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
On Computing the Hermite Form of a Matrix of Differential Polynomials
CASC '09 Proceedings of the 11th International Workshop on Computer Algebra in Scientific Computing
Computing diagonal form and Jacobson normal form of a matrix using Gröbner bases
Journal of Symbolic Computation
Complexity estimates for two uncoupling algorithms
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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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 degx(A), degz(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.