On Serre Reduction of Multidimensional Systems

Abstract

Serre reduction of a system plays a key role in the theory of Multidimensional systems, which has a close connection with Serre reduction of polynomial matrices. In this paper, we investigate the Serre reduction problem for two kinds of nD polynomial matrices. Some new necessary and sufficient conditions about reducing these matrices to their Smith normal forms are obtained. ese conditions can be easily checked by existing Gröbner basis algorithms of polynomial ideals.

Some Further Results on the Reduction of Two‐Dimensional Systems

“…Lemma 2 (see [16]). Let F(z, w), F 1 (z, w), F 2 (z, w) ∈ K l×l [z, w], and F(z, w) � F 1 (z, w)F 2 (z, w).…”

“…Definition 1 (see [16]). Let F(z, w) ∈ K l×m [z, w] (l ≤ m) with rank r and Φ i be a polynomial defined as follows:…”

“…Definition 2 (see [16]). Let F(z, w) ∈ K l×m [z, w] with full row rank; F(z, w) is said to be zero left prime if the l × l minors of F(z, w) have no common zero in K 2 .…”

“…In 2019, Li et al presented some criteria on an l × l bivariate polynomial matrix F(z, w) which is equivalent to the Smith form diag(I l− 1 , detF(z, w)) with detF(z, w) being an irreducible polynomial [11]. ere are also some results on the equivalence of multivariate polynomial matrices in special cases [13,14,16,18,19], among which when an l × l polynomial matrix F(z) which is equivalent to the Smith form diag (I l− 1 , det F(z)) is investigated, such as detF(z) � x 1 − f(x 2 , . .…”