An Algorithm for Computing Greatest Common Right Divisors of Parametric Ore Polynomials

A new algorithm for computing the parametric greatest common right divisor (GCRD) of a set of parametric Ore polynomials is presented in this paper. The algorithm is based on Gröbner bases for modules. Inspired by the resultant theory in Ore polynomial rings, the Sylvester matrix is defined for a set of Ore polynomials. In the case of non-parametric polynomials, the GCRD of Ore polynomials can be obtained by computing the row echelon form of the Sylvester matrix. For the parametric case, the parametric Sylvester matrix is also defined in the paper. Based on this, under the assumption that the specializations commute with the conjugate operator and derivation in the Ore polynomial ring, the parametric GCRD of parametric Ore polynomials can be obtained by computing the Gröbner basis for the module generated by rows of the parametric Sylvester matrix. As a consequence, the algorithm for computing the parametric GCRD is presented in detail and has been implemented in the computer algebra system Singular.