On Serre Reduction of Multidimensional Systems

Dongmei Li,Licui Zheng,Jinwang Liu

doi:10.1155/2020/7435237

Dongmei Li, Licui Zheng + Show 1 more

Open Access

https://doi.org/10.1155/2020/7435237

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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. These conditions can be easily checked by existing Gröbner basis algorithms of polynomial ideals.

Highlights

Multidimensional systems arise naturally in several fields of control, circuits, signals, and network synthesis
For nD (n ≥ 2) case, since this reduction problem is equivalent to a highly difficult problem, the isomorphism of two finitely presented modules, the criteria determining whether arbitrary nD polynomial matrix can be Serre reduced to its Smith normal form has not been presented so far. e reduction for several special classes of polynomial matrices to their Smith normal forms has been investigated
For the polynomial ring K[x1, x2], Lee and Zak [13] presented a necessary condition for a class of 2D polynomial matrices on reducing them to the Smith normal forms

Summary

Introduction

Multidimensional (nD) systems arise naturally in several fields of control, circuits, signals, and network synthesis (see [1,2,3,4,5]). Serre reduction problem aims at simplifying nD systems to some equivalent forms with fewer equations in fewer unknowns, and rewriting these systems such that the key information on them [6], which does not show obviously in the original forms, can be derived from the new equivalent forms more This involves Serre reduction of multivariate (nD) polynomial matrices to some simple equivalent forms, especially their Smith normal forms. They showed that a class of l × l nD polynomial matrices P can be reduced to its Smith normal form diag (Il−1, d) for det P x1 − f(x2, . E results that we mentioned above mainly study polynomial matrices whose reduced matrices correspond to systems which contains only one equation in one unknown with form diag (Il−1, detF(z)), where F(z) is the polynomial matrix corresponds to the given nD system.

Preliminaries and Results

An Example

Conclusions