On Zero Left Prime Factorizations for Matrices over Unique Factorization Domains

Multidimensional linear systems theory has a wide range of applications in circuits, systems, control of networked systems, signal processing, and other areas (see, e.g., [1, 2]). Multivariate polynomial matrix theory is a well-established tool for these systems, since many problems in the analysis and synthesis of control systems can be well solved using multivariate polynomial matrix techniques [1–3]. In recent years, n-D polynomial matrix factorizations have been widely studied [4–10]. In [11, 12], the zero left prime factorization problemwas raised.-is problem has been solved in [4–6]. -e minor left prime factorization problem has been solved in [7, 10]. In the algorithms given in [7, 10], a fitting ideal of some module over the multivariate (n-D) polynomial ring needs to be computed. It is a little complicated. It is well known that a multivariate polynomial ring over a field is a unique factorization domain. -en, the following problem is interesting.


Introduction
Multidimensional linear systems theory has a wide range of applications in circuits, systems, control of networked systems, signal processing, and other areas (see, e.g., [1,2]). Multivariate polynomial matrix theory is a well-established tool for these systems, since many problems in the analysis and synthesis of control systems can be well solved using multivariate polynomial matrix techniques [1][2][3].
It is well known that a multivariate polynomial ring over a field is a unique factorization domain. en, the following problem is interesting. Problem 1. How to decide if a matrix with full row rank over a unique factorization domain has a zero left prime factorization?
In this paper, we will give a partial solution to this problem.

Preliminaries
Let R be a unique factorization domain. e set of all l × m matrices with entries from R is denoted by R l×m . Let F ∈ R l×m (l < m). We denote the greatest common divisor of all l × l minors of F by d(F). Let C ∈ R l×l be a submatrix of F. By deleting C from F, we get a submatrix of F. is submatrix is denoted by F\C. Let C ∈ R m×m . adj(C) denotes the adjoint matrix of C. acof ij (C) denotes the i, jth algebraic cofactor of C.
, and let C ∈ R l×l be a submatrix of F. A minor of F consisting of l − 1 columns from C and one column from F\C is said to be a related minor of C. e following definition is from the multidimensional systems theory [13].
where C ∈ R l×l and F 1 ∈ R l×m . If F 1 is ZLP, then this factorization is said to be a zero left prime factorization.

Main Results
First, we need a lemma.
where C ∈ R l×l and C ∈ R l×(m− l) . en, the elements of adj C · C are just all related minors of C (up to a sign).
by Laplace eorem. us, b ij is a related minor of C (up to a sign). It is clear that they are just all related minors of C (up to a sign). Now, we prove the main theorem of this paper.
□ Theorem 1. Let F ∈ R l×m (l < m). If there exists an l × l submatrix C of F such that detC is a common factor of all related minors of C, then there exists F 1 ∈ R l×m such that F � CF 1 and F 1 is ZLP; i.e., F has a ZLP factorization.
Proof. We can change the order of the columns of F such that the submatrix C consists of the left l columns of F. us, there exists an invertible matrix Q ∈ R m×m such that FQ � (C, C), where C ∈ R l×l and C ∈ R l×(m− l) . Since detC is a common factor of all related minors of C, by Lemma 1, en, Q 1 ∈ R m×m . We have en, (4) If there exists an l × l submatrix C of F such that detC is a common factor of all related minors of C, then detC � d(F).
Proof. Clearly, d(F) | detC. By eorem 1, there exists F 1 ∈ R l×m such that F � CF 1 . By Cauchy-Binet formula, we have detC | d(F). erefore, detC � d(F). □ Corollary 2. Let F ∈ R l×m (l < m). If there exists an l × l submatrix C of F such that detC is a common factor of all related minors of C, then F is equivalent to (C, O).

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this article.