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

  • Mark Giesbrecht
  • Albert Heinle
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7442)

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.

Keywords

Normal Form Hermite Form Degree Bound Principal Ideal Domain Unimodular Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Mark Giesbrecht
    • 1
  • Albert Heinle
    • 2
  1. 1.Cheriton School of Computer ScienceUniversity of WaterlooCanada
  2. 2.Lehrstuhl D für MathematikRWTH Aachen UniversityAachenGermany

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