Smith Form Equivalence for Several Classes of Multivariate Polynomial Matrices

TL;DR

This paper studies the conditions under which multivariate polynomial matrices can be simplified to Smith form, using algebraic tools and algorithmic methods for verification.

Contribution

It introduces criteria for Smith form equivalence of polynomial matrices and extends these results to non-square and rank-deficient cases using algebraic theorems.

Findings

Criteria for Smith form equivalence established

Extension to non-square and rank-deficient matrices achieved

Algorithmic verification via Gröbner basis methods possible

Abstract

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.