Asymptotically fast triangularization of matrices over rings

Reviewer: H. G. Zimmer

The authors propose two deterministic algorithms for computing the Hermite normal form and the Smith normal form of an integer nonsquare matrix. Their approach is an extension of corresponding algorithms designed by Domich, Kannan, Trotter, and Iliopolous for square integer matrices. A complexity analysis of the new algorithms also yields their running times. The growth of intermediate expressions, which occurs in the classical Euclidean procedure, is circumvented here by the use of modular arithmetic. In the main part of the paper, fast matrix multiplication techniques are applied to the triangularization problem of matrices over principal ideal rings (PIRs) and principal ideal domains (PIDs) by means of elementary column operations, and the number of ring operations the algorithms require is estimated. Finally, these techniques are used in a triangularization algorithm over the ring of integers. This last algorithm turns out to be faster than the Hermite normal form algorithm but does not produce the unique Hermite normal form. The triangular matrix obtained has small entries (which are estimated in the paper) similar to the Hermite normal form, however, and can therefore serve as a suitable substitute. In a concluding section, a fast probabilistic algorithm for computing the Smith normal form, based on a method of Kaltofen, Krishnamoorthy, and Saunders, is described, and some open problems are mentioned, such as the problem of detecting wrong answers of the probabilistic algorithm. One should expect these asymptotically fast polynomial-time triangularization algorithms to b e fast in practice too, but apparently no practical tests have been carried through. This long but well-written paper contains a wealth of interesting material having applications to such areas as lattice theory, linear Diophantine equations, the theory of modules over PIDs, and integer programming.