A Fast Las Vegas Algorithm for Computing the Smith Normal Form of a Polynomial Matrix

A Las Vegas probabilistic algorithm is presented that finds the Smith normal form S ∈ Q[x] of a nonsingular input matrix A ∈ Z [x]. The algorithm requires an expected number of O (̃nd(d + n log ||A||)) bit operations (where ||A|| bounds the magnitude of all integer coefficients appearing in A and d bounds the degrees of entries of A). In practice, the main cost of the computation is obtaining a non-unimodular triangularization of a polynomial matrix of same dimension and with similar size entries as the input matrix. We show how to accomplish this in O (̃nd(d + log ||A||) log ||A||) bit operations using standard integer, polynomial and matrix arithmetic. These complexity results improve significantly on previous algorithms in both a theoretical and practical sense.