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 Grobner 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.& COPY; 2023 Elsevier Ltd. All rights reserved.

Automated summary

This paper investigates the reduction of weakly linear multivariate polynomial matrices to their Smith normal forms, using hierarchical-recursive method and Quillen-Suslin Theorem. The necessary and sufficient conditions for such matrices to be reduced to their Smith normal forms are derived, which can be easily checked by computing the reduced Grobner bases of the relevant polynomial ideals. Based on the new results, an algorithm for reducing weakly linear multivariate polynomial matrices to their Smith normal forms is proposed.