Output-sensitive modular algorithms for polynomial matrix normal forms

We give modular algorithms to compute row-reduced forms, weak Popov forms, and Popov forms of polynomial matrices, as well as the corresponding unimodular transformation matrices. Our algorithms improve on existing fraction-free algorithms. In each case we define lucky homomorphisms, determine the appropriate normalization, as well as bound the number of homomorphic images required. The algorithms have the advantage that they are output-sensitive, that is, the number of homomorphic images required depends on the size of the output. Furthermore, there is no need to verify the result by trial division or multiplication. Our algorithms can be used to compute normalized one-sided greatest common divisors and least common multiples of polynomial matrices along with irreducible matrix-fraction descriptions of matrix rational functions. When our algorithm is used to compute polynomial greatest common divisors, we obtain a new output-sensitive modular algorithm.