Smith Form Equivalence for Several Classes of Multivariate Polynomial Matrices

This paper investigates the equivalence reduction for several classes of multivariate polynomial matrices and their Smith forms, establishing some criteria for such reduction. In particular, we employ algebra isomorphisms as a key tool to study this equivalence problem. We then leverage the Quillen-Suslin and Lin-Bose theorems to extend these results to non-square and rank-deficient matrices. Moreover, the verification of our criteria is achievable algorithmically via existing Gr"{o}bner basis methods.