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The issue of the semiscalar equivalence of Laurent polynomial matrices is investigated and the triangular form of such matrices and their finite sets is established with respect to this equivalence.  The theorem on regularization of a Laurent polynomial matrix is proved.  This theorem is used in the problem of factorization of such matrices.  The factorization criterion of a Laurent polynomial matrix with a regular multiplier with a predetermined Smith normal form is obtained.

Triangular form of Laurent polynomial matrices and their factorization