Algorithms for Normal Forms for Matrices of Polynomials and Ore Polynomials

I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii In this thesis we study algorithms for computing normal forms for matrices of Ore polynomials while controlling coefficient growth. By formulating row reduction as a linear algebra problem, we obtain a fraction-free algorithm for row reduction for matrices of Ore polynomials. The algorithm allows us to compute the rank and a basis of the left nullspace of the input matrix. When the input is restricted to matrices of shift polynomials and ordinary polynomials, we obtain fraction-free algorithms for computing row-reduced forms and weak Popov forms. These algorithms can be used to compute a greatest common right divisor and a least common left multiple of such matrices. Our fraction-free row reduction algorithm can be viewed as a generalization of subresultant algorithms. The linear algebra formulation allows us to obtain bounds on the size of the intermediate results and to analyze the complexity of our algorithms. We then make use of the fraction-free algorithm as a basis to formulate modular algorithms for computing a row-reduced form, a weak Popov form, and the Popov form of a polynomial matrix. By examining the linear algebra formulation, we develop criteria for detecting unlucky homomorphisms and determining the number of homomorphic images required. iii Acknowledgements I wish to thank my supervisor, George Labahn, for his support for the duration of this work. Without his suggestions, criticisms, friendship, and financial support, it would not have been possible to complete this work. I also learned much from Bern-hard Beckermann when we worked together to obtain some of the results reported in this thesis. I also wish to thank Dr. Keith Geddes, Dr. Mark Giesbrecht, Dr. David Saunders, and Dr. Cameron Stewart for serving on the thesis committee