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The equivalence of multidimensional systems is closely related to the equivalence of multivariate polynomial matrices, for which the Smith form plays an important role. In this paper we study multivariate polynomial matrices with their entries in the polynomial ring, whereis an arbitrary field. We derive some new conditions on reducing these matrices to their Smith forms. These conditions can be verified by computing the reduced Gröbner bases of the associated ideals.

The smith form of a multivariate polynomial matrix over an arbitrary coefficient field