On the equivalence of multivariate polynomial matrices

Serre’s Reduction and the Smith Forms of Multivariate Polynomial Matrices

“…It also gives a sufficient condition to check this kind of matrices are equivalent to their Smith forms; in fact, this condition is also a necessary condition. When K � R is the real field and q 2 � 0, eorem 4 is same as eorem 2.5 in Li et al [17]. Furthermore, if q 1 � 1 and q 2 � 0, eorem 4 is just Proposition 4 in Lin et al [12].…”

“…Proof. e proof is similar to that of Lemma 2.2 in [17], so it is omitted here. □ Lemma 4 (see [18]).…”

“…Definition 3 (see [17]). Let T 1 (z) and T 2 (z) denote two matrices in K l×m [z], T 1 (z), and T 2 (z) are said to be equivalent if there exist two invertible matrices M(z) ∈ K l×l [z] and…”

“…, z n ) and proved that F(z) is equivalent to its Smith form. Li et al [17] have investigated Problem 1 for F(z) ∈ R l×l [z] with detF(z) � (z 1 − f(z 2 , . .…”

On Serre Reduction of Multidimensional Systems

“…Furthermore, they showed that a class of l × l nD polynomial matrices P can be reduced to its Smith normal form diag Boudellioua [15] gave some similar conditions using module theoretic approach. Li et al [16] proposed a sufficient and necessary condition under which P can be reduced to its Smith normal form diag…”