On the recursive equivalence to Smith form of multivariate polynomial matrices

Abstract

Abstract The Smith form of an $n$-$D$ polynomial matrix plays an important role in many areas of mathematics and engineering. In this paper, we investigate the recursive equivalence problem of $n$-dimensional polynomial matrices, i.e. if diag$(1,B)$ is equivalent to diag$(1,1,C)$, is B equivalent to diag$(1,C)$? We give a negative answer to this question by explicitly constructing a four-dimensional polynomial matrix which is not equivalent to its Smith form.