The Smith normal form and reduction of weakly linear matrices.

The reduction of a multidimensional system is closely related to the reduction of a multivariate polynomial matrix, for which the Smith normal form of the matrix plays a key role. In this paper, we investigate the reduction of weakly linear multivariate polynomial matrices to their Smith normal forms. Using hierarchical-recursive method and Quillen-Suslin Theorem, we derive some necessary and sufficient conditions ensuring that such matrices can be reduced to their Smith normal forms, and these conditions are easily checked by computing the reduced Gröbner bases of the relevant polynomial ideals. Based on the new results, we propose an algorithm for reducing weakly linear multivariate polynomial matrices to their Smith normal forms. [ABSTRACT FROM AUTHOR]