New Remarks on the Factorization and Equivalence Problems for a Class of Multivariate Polynomial Matrices
aa r X i v : . [ c s . S C ] O c t New Remarks on the Factorization and Equivalence Problems fora Class of Multivariate Polynomial Matrices
Dong Lu a,b , Dingkang Wang c,d , Fanghui Xiao e, ∗ a Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing 100191, China b School of Mathematical Sciences, Beihang University, Beijing 100191, China c KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China d School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China e College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua, 321004, China
Abstract
This paper is concerned with the factorization and equivalence problems of multivariate poly-nomial matrices. We present some new criteria for the existence of matrix factorizations for aclass of multivariate polynomial matrices, and obtain a necessary and su ffi cient condition forthe equivalence of a square polynomial matrix and a diagonal matrix. Based on the construc-tive proof of the new criteria, we give a factorization algorithm and prove the uniqueness of thefactorization. We implement the algorithm on Maple, and two illustrative examples are given toshow the e ff ectiveness of the algorithm. Keywords:
Multivariate polynomial matrices, Matrix factorization, Matrix equivalence,Column reduced minors, Gr¨obner basis
1. Introduction
Multidimensional systems have wide applications in image, signal processing, control of net-worked systems, and other areas (see, e.g., Bose (1982); Bose et al. (2003)). A multidimensionalsystem may be represented by a multivariate polynomial matrix, and we can obtain some impor-tant properties of the system by studying the corresponding matrix. Symbolic computation pro-vides many e ff ective theories and algorithms, such as module theory and Gr¨obner basis algorithm(Cox et al., 2005; Lin et al., 2008), for the research of multidimensional systems. Therefore, thefactorization and equivalence problems related to multivariate polynomial matrices have madegreat progress over the past decades.Up to now, the factorization problem for univariate and bivariate polynomial matrices hasbeen completely solved by Morf et al. (1977); Guiver and Bose (1982); Liu and Wang (2013),but the case of more than two variables is still open. Youla and Gnavi (1979) first introducedthree important concepts according to di ff erent properties of multivariate polynomial matrices,namely zero prime matrix factorization, minor prime matrix factorization and factor prime ma-trix factorization. When multivariate polynomial matrices satisfy several special properties, there ∗ Corresponding author
Email addresses: [email protected] (Dong Lu), [email protected] (Dingkang Wang), [email protected] (Fanghui Xiao) re some results about the existence problem of zero prime matrix factorizations for the matrices(see, e.g., Charoenlarpnopparut and Bose (1999); Lin (1999a, 2001)). After that, Lin and Bose(2001) proposed the famous Lin-Bose conjecture: a multivariate polynomial matrix admits a zeroprime matrix factorization if all its maximal reduced minors generate a unit ideal. This conjec-ture was proved by Pommaret (2001); Srinivas (2004); Wang and Feng (2004); Liu et al. (2014),respectively. Wang and Kwong (2005) gave a necessary and su ffi cient condition for a multivari-ate polynomial matrix with full rank to have a minor prime matrix factorization. They extractedan algorithm from Pommaret’s proof of the Lin-Bose conjecture, and examples showed the e ff ec-tiveness of the algorithm. Guan et al. (2019) generalized the main results in Wang and Kwong(2005) to the case of multivariate polynomial matrices without full rank. For the existence prob-lem of factor prime matrix factorizations for multivariate polynomial matrices with full rank,Wang (2007) and Liu and Wang (2010) introduced the concept of regularity and obtained a nec-essary and su ffi cient condition. Guan et al. (2018) gave an algorithm to judge whether a mul-tivariate polynomial matrix with the greatest common divisor of all its maximal minors beingsquare-free has a factor prime matrix factorization. However, the existence problem for fac-tor prime matrix factorizations of multivariate polynomial matrices remains a challenging openproblem so far.Comparing to the factorization problem of multivariate polynomial matrices which has beenwidely investigated during the past years, less attention has been paid to the equivalence problemof multivariate polynomial matrices. For any given multidimensional system, our goal is tosimplify it into a simpler equivalent form.Since a univariate polynomial ring is a principal ideal domain, a univariate polynomial matrixis always equivalent to its Smith form. This implies that the equivalence problem of univariatepolynomial matrices has been solved (see, e.g., Rosenbrock (1970); Kailath (1993)). For anygiven bivariate polynomial matrix, conditions under which it is equivalent to its Smith formhave been investigated by Frost and Storey (1978); Lee and Zak (1983); Frost and Boudellioua(1986). Note that the equivalence problem of two multivariate polynomial matrices is equivalentto the isomorphism problem of two finitely presented modules. Boudellioua and Quadrat (2010)and Cluzeau and Quadrat (2008, 2013, 2015) obtained some important results by using moduletheory and homological algebra. According to the works of Boudellioua and Quadrat (2010),Boudellioua (2012, 2014) designed some algorithms based on Maple to compute Smith formsfor some classes of multivariate polynomial matrices. For the case of multivariate polynomialmatrices with more than one variable, however, the equivalence problem is not yet fully solveddue to the lack of a mature polynomial matrix theory (see, e.g., Kung et al. (1977); Morf et al.(1977); Pugh et al. (1998)).From our personal viewpoint, new ideas need to be injected into these areas to obtain newtheoretical results and e ff ective algorithms. Therefore, it would be significant to provide somenew criteria to study the factorization and equivalence problems for some classes of multivariatepolynomial matrices.From the 1990s to the present, there is a class of multivariate polynomial matrices that hasalways attracted attention. That is, M = { F ∈ k [ z ] l × m : z − f ( z ) is a divisor of d l ( F ) with f ( z ) ∈ k [ z ] } , where l ≤ m , z = { z , . . . , z n } with n ≥ z = { z , . . . , z n } and d l ( F ) is the greatest commondivisor of all the l × l minors of F . People tried to solve the factorization and equivalence problemsof multivariate polynomial matrices in M . Let F ∈ M and h = z − f ( z ). Many factorization2riteria on the existence of a matrix factorization for F with respect to h have been proposed(see, e.g., Lin et al. (2001); Liu et al. (2011); Lu et al. (2020a)). When l = m and det( F ) = h ,Lin et al. (2006) proved that F is equivalent to its Smith form. After that, Li et al. (2017) studiedthe equivalence problem of a square matrix F with det( F ) = h r and a diagonal matrix, where r ≥ M without sat-isfying previous factorization criteria or equivalence conditions, but they can be factorized withrespect to h or equivalent to simpler forms. As a consequence, we continue to study the factor-ization and equivalence problems of multivariate polynomial matrices in M .This paper is an extension of Lu et al. (2020b), and the contributions listed following arenew. 1) Under the assumption that h is not a divisor of the greatest common divisor of all the( l − × ( l −
1) minors of F , we give a necessary and su ffi cient condition for the existence ofa matrix factorization of F with respect to h . 2) We summarize all factorization criteria for theexistence of a matrix factorization of F with respect to h , and study the relationships amongthem. 3) For the case that h is a divisor of the greatest common divisor of all the ( l − × ( l − F , we obtain a su ffi cient condition for the existence of a matrix factorization of F with respect to h r , where 2 ≤ r ≤ l . 4) Based on the new factorization criteria, we construct anew factorization algorithm and implement it on Maple; codes and examples are available on thewebsite: .The rest of the paper is organized as follows. After a brief introduction to matrix factorizationand matrix equivalence in section 2, we use two examples to propose two problems that we shallconsider. We present in section 3 two criteria for factorizing F with respect to h , and then studythe relationships among all existed factorization criteria. A necessary and su ffi cient conditionfor the equivalence of a square polynomial matrix and a diagonal matrix is described in section4. We in section 5 generalize the main result in section 3 to a more general case. We in section6 construct a factorization algorithm and study the uniqueness of matrix factorizations by thealgorithm, and use two examples to illustrate the e ff ectiveness of the algorithm in section 7. Thepaper contains a summary of contributions and some remarks in section 8.
2. Preliminaries and problems
In this section we first recall some basic notions which will be used in the following sections,and then we use two examples to put forward two problems that we are considering.
We denote by k an algebraically closed field. Let k [ z ] and k [ z ] be the polynomial ring invariables z and z with coe ffi cients in k , respectively. Let k [ z ] l × m be the set of l × m matrices withentries in k [ z ]. Throughout the paper, we assume that l ≤ m , and use uppercase bold letters todenote polynomial matrices. In addition, “w.r.t.” stands for “with respect to”.Let F ∈ k [ z ] l × m , we use d i ( F ) to denote the greatest common divisor of all the i × i minors of F with the convention that d ( F ) =
1, where i = , . . . , l . Let f ∈ k [ z ], then F ( f , z ) denotes apolynomial matrix in k [ z ] l × m which is formed by transforming z in F into f . Definition 1 (Lin (1988); Sule (1994)) . Let F ∈ k [ z ] l × m with rank r, where ≤ r ≤ l. For anygiven integer i with ≤ i ≤ r, let a , . . . , a β denote all the i × i minors of F , where β = (cid:16) li (cid:17) · (cid:16) mi (cid:17) .Extracting d i ( F ) from a , . . . , a β yieldsa j = d i ( F ) · b j , j = , . . . , β, hen b , . . . , b β are called the i × i reduced minors of F . Lin (1988) showed that reduced minors are important invariants for polynomial matrices.
Lemma 2.
Let F ∈ k [ z ] r × t be of full row rank, b , . . . , b γ be all the r × r reduced minors of F ,and F ∈ k [ z ] t × ( t − r ) be of full column rank, ¯ b , . . . , ¯ b γ be all the ( t − r ) × ( t − r ) reduced minors of F , where r < t and γ = (cid:16) tr (cid:17) . If F F = r × ( t − r ) , then ¯ b i = ± b i for i = , . . . , γ , and signs dependon indices. Let F ∈ k [ z ] l × m with rank r , where 1 ≤ r ≤ l . Let ¯ F , . . . , ¯ F η ∈ k [ z ] l × r be all the fullcolumn rank submatrices of F , where 1 ≤ η ≤ (cid:16) mr (cid:17) . According to Lemma 2, it is easy to provethat ¯ F , . . . , ¯ F η have the same r × r reduced minors. Based on this phenomenon, we give thefollowing concept. Definition 3.
Let F ∈ k [ z ] l × m with rank r, and ¯ F ∈ k [ z ] l × r be an arbitrary full column ranksubmatrix of F , where ≤ r ≤ l. Let c , . . . , c ξ be all the r × r reduced minors of ¯ F , where ξ = (cid:16) lr (cid:17) .Then c , . . . , c ξ are called the r × r column reduced minors of F . We can define the r × r row reduced minors of F in the same way.In order to state conveniently problems and main conclusions of this paper, we introduce thefollowing concepts and results. Definition 4.
Let F ∈ k [ z ] l × m be of full row rank. If all the l × l minors of F generate k [ z ] , then F is said to be a zero left prime (ZLP) matrix. If all the l × l minors of F are relatively prime, i.e., d l ( F ) is a nonzero constant in k, then F is said to be a minor left prime (MLP) matrix. If for any polynomial matrix factorization F = F F with F ∈ k [ z ] l × l , F is necessarily aunimodular matrix, i.e., det( F ) is a nonzero constant in k, then F is said to be a factor leftprime (FLP) matrix. Zero right prime (ZRP) matrices, minor right prime (MRP) matrices and factor right prime(FRP) matrices can be similarly defined for matrices F ∈ k [ z ] m × l with m ≥ l . We refer toYoula and Gnavi (1979) for more details about the relationships among ZLP matrices, MLP ma-trices and FLP matrices.For any given ZLP matrix F ∈ k [ z ] l × m , Quillen (1976) and Suslin (1976) proved that an m × m unimodular matrix can be constructed such that F is its first l rows, respectively. This result iscalled Quillen-Suslin theorem, and it solved the problem raised by Serre (1955). Lemma 5. If F ∈ k [ z ] l × m is a ZLP matrix, then a unimodular matrix U ∈ k [ z ] m × m can beconstructed such that F is its first l rows. There are many algorithms for the Quillen-Suslin theorem, we refer to Youla and Pickel(1984); Logar and Sturmfels (1992); Park (1995) for more details. Fabia´nska and Quadrat (2007)first designed a Maple package, which is called QUILLENSUSLIN, to implement the Quillen-Suslin theorem.Let W be a k [ z ]-module generated by ~ u , . . . , ~ u l ∈ k [ z ] × m . The set of all ( b , . . . , b l ) ∈ k [ z ] × l such that b ~ u + · · · + b l ~ u l = ~ k [ z ]-module of k [ z ] × l , is called the (first) syzygy moduleof W , and denoted by Syz( W ). Lin (1999b) proposed several interesting structural propertiesof syzygy modules. Let F = h ~ u T1 , . . . , ~ u T l i T . The rank of W is defined as the rank of F that isdenoted by rank( F ). Guan et al. (2018) proved that the rank of W does not depend on the choiceof generators of W . 4 emma 6. With above notations. If rank( W ) = r with ≤ r ≤ l, then the rank of Syz( W ) is l − r.Proof. Let k ( z ) be the fraction field of k [ z ], and Syz ∗ ( W ) = { ~ v ∈ k ( z ) × l : ~ v · F = ~ } . Then,Syz ∗ ( W ) is a k ( z )-vector space of dimension l − r . For any given l − r + ff erent vectors ~ v , . . . ,~ v l − r + ∈ k [ z ] × l in Syz( W ), it is obvious that ~ v i ∈ Syz ∗ ( W ) for each i , and they are k ( z )-linearlydependent. This implies that ~ v , . . . ,~ v l − r + are k [ z ]-linearly dependent. Thus rank(Syz( W )) ≤ l − r .Assume that ~ p , . . . , ~ p l − r ∈ k ( z ) × l are l − r vectors in Syz ∗ ( W ), and they are k ( z )-linearlyindependent. For each j , we have p j ~ u + · · · + p jl ~ u l = ~
0, where ~ p j = ( p j , . . . , p jl ). Multiplyingboth sides of the equation by the least common multiple of the denominators of p j , . . . , p jl , weobtain ¯ p j = ( ¯ p j , . . . , ¯ p jl ) ∈ k [ z ] such that ¯ p j ~ u + · · · + ¯ p jl ~ u l = ~
0. Then, ¯ p j ∈ Syz( W ), where j = , . . . , l − r . Moreover, ¯ p , . . . , ¯ p l − r are k [ z ]-linearly independent. Thus, rank(Syz( W )) ≥ l − r .As a consequence, the rank of Syz( W ) is l − r and the proof is completed. Remark 7.
Assume that
Syz( W ) is generated by ~ v , . . . ,~ v t ∈ k [ z ] × l , and H = h ~ v T1 , . . . ,~ v T t i T . Itfollows from rank( H ) = l − r that t ≥ l − r. That is, the number of vectors in any given generatorsof Syz( W ) is greater than or equal to l − r. Let F ∈ k [ z ] l × m with rank r , where 1 ≤ r ≤ l . For each 1 ≤ i ≤ r , we use I i ( F ) to denotethe ideal generated by all the i × i minors of F . For convenience, let I ( F ) = k [ z ]. Moreover, wedenote the submodule of k [ z ] × m generated by all the row vectors of F by Im( F ). Definition 8.
Let W be a finitely generated k [ z ] -module, and k [ z ] × l φ −−→ k [ z ] × m → W → be apresentation of W, where φ acts on the right on row vectors, i.e., φ ( ~ u ) = ~ u · F for ~ u ∈ k [ z ] × l with F being a presentation matrix corresponding to the linear mapping φ . Then the ideal Fitt j ( W ) = I m − j ( F ) is called the j-th Fitting ideal of W. Here, we make the convention that Fitt j ( W ) = k [ z ] for j ≥ m, and that Fitt j ( W ) = for j < max { m − l , } . We remark that
Fitt j ( W ) only depend on W (see, e.g., Greuel and Pfister (2002); Eisenbud(2013)). In addition, the chain 0 = Fitt − ( W ) ⊆ Fitt ( W ) ⊆ . . . ⊆ Fitt m ( W ) = k [ z ] of Fittingideals is increasing. Cox et al. (2005) showed that one obtains the presentation matrix F for W by arranging the generators of Syz( W ) as rows. A matrix factorization of a multivariate polynomial matrix is formulated as follows.
Definition 9.
Let F ∈ k [ z ] l × m and h | d l ( F ) . F is said to admit a matrix factorization w.r.t. h if F can be factorized as F = G F (1) such that G ∈ k [ z ] l × l with det( G ) = h and F ∈ k [ z ] l × m . In particular, Equation (1) is said tobe a ZLP (MLP, FLP) matrix factorization if F is a ZLP (MLP, FLP) matrix. Throughout the paper, let h = z − f ( z ) with f ( z ) ∈ k [ z ]. This paper will address thefollowing specific matrix factorization problem. Problem 10.
Let F ∈ M . Under what conditions do F have a matrix factorization w.r.t. h. So far, several results have been made on Problem 10, and the latest progress on this problemwas obtained by Lu et al. (2020a). 5 emma 11.
Let F ∈ M . If h ∤ d l − ( F ) and the ideal generated by h and all the ( l − × ( l − reduced minors of F is k [ z ] , then F admits a matrix factorization w.r.t. h. Although Lemma 11 gives a criterion to determine whether F has a matrix factorization w.r.t. h , we found that there exist some polynomial matrices in M which do not satisfy the conditionsof Lemma 11, but still admit matrix factorizations w.r.t. h . Example 12.
Let F = " − z z + z z + z z − z z + z z z − z − z z + z z z z − z z z − z z + z − z + z z z be a polynomial matrix in C [ z , z , z ] × , where C is the complex field.It is easy to compute that d ( F ) = z ( z − z ) and d ( F ) = . Let h = z − z , then h | d ( F ) implies that F ∈ M . Obviously, h ∤ d ( F ) . Since d ( F ) = , the entries in F are all the × reduced minors of F . Let ≺ z be the degree reverse lexicographic order, then the reduced Gr¨obnerbasis G of the ideal generated by h and all the × reduced minors of F w.r.t. ≺ z is { z − z , z , z } .It follows from G , { } that Lemma 11 cannot be applied.However, F admits a matrix factorization w.r.t. h, i.e., there exist G ∈ C [ z , z , z ] × and F ∈ C [ z , z , z ] × such that F = G F = " h z z z − z z − z z z − z z + z − z + z z z , where det( G ) = h. From the above example we see that Problem 10 is far from being resolved. So, in the nextsection we make a detailed analysis on this problem.
Now we introduce the concept of the equivalence of two multivariate polynomial matrices.
Definition 13.
Two polynomial matrices F ∈ k [ z ] l × m and F ∈ k [ z ] l × m are said to be equivalentif there exist two unimodular matrices U ∈ k [ z ] l × l and V ∈ k [ z ] m × m such that F = UF V . (2)In fact, a univariate polynomial matrix is always equivalent to its Smith form. However, thisresult is not valid for the case of more than one variable, and there are many counter-examples(see, e.g., Lee and Zak (1983); Boudellioua (2013)). Hence, people began to consider under whatconditions multivariate polynomial matrices in k [ z ] are equivalent to simpler forms. Li et al.(2017) investigated the equivalence problem for a class of multivariate polynomial matrices andobtained the following result. Lemma 14.
Let F ∈ k [ z ] l × l with det( F ) = h r , where h = z − f ( z ) and r is a positive integer.Then F is equivalent to diag( h r , , . . . , if and only if h r and all the ( l − × ( l − minors of F generate k [ z ] . For a given square matrix that does not satisfy the condition of Lemma 14, we use the fol-lowing example to illustrate that it can be equivalent to another diagonal matrix.6 xample 15.
Let F = z z − z + z z + z − z − z z z − z z + z z − z + z z − z z z z − z z z z − z + z − z + z + z − z )( z z + z + z + + z F [2 , z − z z z − z z + z − z z z − z z + z − z be a polynomial matrix in C [ z , z , z ] × , where F [2 , = z z z − z z + z z − z + z z − z z ,and C is the complex field.It is easy to compute that det( F ) = ( z − z ) . Let h = z − z and ≺ z be the degree reverselexicographic order, then the reduced Gr¨obner basis G of the ideal generated by h and all the × minors of F w.r.t. ≺ z is { z − z } . It follows from G , { } that Lemma 14 cannot be applied.However, F is equivalent to diag( h , h , , i.e., there exist two unimodular polynomial matrices U ∈ C [ z , z , z ] × and V ∈ C [ z , z , z ] × such that F = U · diag( h , h , · V = z z − z z + h h
00 0 1 z + z z + z . Based on the phenomenon of Example 15, we consider the following matrix equivalenceproblem in this paper.
Problem 16.
Let F ∈ k [ z ] l × l with det( F ) = h r , where h = z − f ( z ) and ≤ r ≤ l. What is thesu ffi cient and necessary condition for the equivalence of F and diag( h , . . . , h | {z } r , , . . . , | {z } l − r ) ?
3. Factorization for polynomial matrices
In this section, we first propose two criteria to judge whether F ∈ M has a matrix factorizationw.r.t. h , and then study the relationships among all existed factorization criteria. ffi cient condition We first introduce two lemmas.
Lemma 17 (Wang and Feng (2004)) . Let F ∈ k [ z ] l × m with rank r, and all the r × r reducedminors of F generate k [ z ] . Then there exist G ∈ k [ z ] l × r and F ∈ k [ z ] r × m such that F = G F with F being a ZLP matrix. Lemma 18 (Lin et al. (2001)) . Let p ∈ k [ z ] and f ( z ) ∈ k [ z ] . Then z − f ( z ) is a divisor of pif and only if p ( f , z ) is a zero polynomial in k [ z ] . Now, we propose a su ffi cient condition to factorize F w.r.t. h . Theorem 19.
Let F ∈ M and W = Im( F ( f , z )) . If Fitt l − ( W ) = and Fitt l − ( W ) = h d i withd ∈ k [ z ] \ { } , then F admits a matrix factorization w.r.t. h.Proof. Let k [ z ] × s φ −−→ k [ z ] × l → W → W , and H ∈ k [ z ] s × l be a matrixcorresponding to the linear mapping φ . Then Syz( W ) = Im( H ).It follows from Fitt l − ( W ) = × H are zero polynomials. Then,rank( H ) ≤
1. Moreover,
Fitt l − ( W ) = h d i with d ∈ k [ z ] \ { } implies that rank( H ) ≥
1. As aconsequence, we have rank( H ) =
1. 7et a , . . . , a β ∈ k [ z ] and b , . . . , b β ∈ k [ z ] be all the 1 × × H , respectively. Then, a i = d ( H ) · b i for i = , . . . , β . Since h a , . . . , a β i = h d i , it is obviousthat d | d ( H ). Moreover, we have d = P β i = c i a i for some c i ∈ k [ z ]. Thus d = d ( H ) · ( P β i = c i b i ).This implies that d ( H ) | d . Hence d = δ · d ( H ), where δ is a nonzero constant. Therefore, h b , . . . , b β i = k [ z ].According to Lemma 17, there exist ~ u ∈ k [ z ] s × and ~ w ∈ k [ z ] × l such that H = ~ u ~ w with ~ w being a ZLP vector. It follows from Syz( W ) = Im( H ) that ~ u ~ w F ( f , z ) = s × m . Since ~ u is acolumn vector, we have ~ w F ( f , z ) = × m .Using the Quillen-Suslin theorem, we can construct a unimodular matrix U ∈ k [ z ] l × l suchthat ~ w is its first row. Let F = UF , then the first row of F ( f , z ) = UF ( f , z ) is zero vector. ByLemma 18, h is a common divisor of the polynomials in the first row of F , thus F = UF = DF = diag( h , , . . . , | {z } l − ) · ¯ f ¯ f · · · ¯ f m ... ... ... ... ¯ f l ¯ f l · · · ¯ f lm . Consequently, we can now derive the matrix factorization of F w.r.t. h , i.e., F = G F , where G = U − D ∈ k [ z ] l × l , F ∈ k [ z ] l × m and det( G ) = h . ffi cient condition for a special case In Theorem 19, the conditions
Fitt l − ( W ) = Fitt l − ( W ) = h d i imply that the rankof F ( f , z ) is l −
1. In the following, we first give a lemma about the necessary and su ffi cientcondition for rank( F ( f , z )) = l − Lemma 20.
Let F ∈ M . Then rank( F ( f , z )) = l − if and only if h ∤ d l − ( F ) .Proof. Since h | d l ( F ), we have rank( F ( f , z )) ≤ l −
1. Let a , . . . , a γ ∈ k [ z ] be all the ( l − × ( l − F , then a ( f , z ) , . . . , a γ ( f , z ) are all the ( l − × ( l −
1) minors of F ( f , z ).Assume that rank( F ( f , z )) = l −
1, then there is at least one integer i with 1 ≤ i ≤ γ such that a i ( f , z ) is a nonzero polynomial. According to Lemma 18, h is not a divisor of a i . Obviously, h ∤ d l − ( F ).Suppose h ∤ d l − ( F ). If rank( F ( f , z )) < l −
1, then a j ( f , z ) = j = , . . . , γ . Thisimplies that h is a common divisor of a , . . . , a γ , which leads to a contradiction. Therefore,rank( F ( f , z )) = l − Lemma 21 (Lin et al. (2005)) . Let G ∈ k [ z ] l × l with det( G ) = h, then there is a ZLP vector ~ w ∈ k [ z ] × l such that ~ w G ( f , z ) = × l . Now, we give a partial solution to Problem 10.
Theorem 22.
Let F ∈ M with h ∤ d l − ( F ) . Then the following are equivalent: F admits a matrix factorization w.r.t. h; all the ( l − × ( l − column reduced minors of F ( f , z ) generate k [ z ] .Proof. →
2. If F admits a matrix factorization w.r.t. h , then there are G ∈ k [ z ] l × l and F ∈ k [ z ] l × m such that F = G F with det( G ) = h . Obviously, F ( f , z ) = G ( f , z ) F ( f , z ).Since det( G ) = h , by Lemma 21 there is a ZLP vector ~ w ∈ k [ z ] × l such that ~ w G ( f , z ) = × l .This implies that ~ w F ( f , z ) = × m . According to Lemma 20, we have rank( F ( f , z )) = l − l − × ( l −
1) column reduced minors of F ( f , z ) are equivalent to allthe 1 × ~ w . It follows that all the ( l − × ( l −
1) column reduced minors of F ( f , z ) generate k [ z ].2 →
1. Sine rank( F ( f , z )) = l −
1, there is a nonzero vector ~ w = [ w , . . . , w l ] ∈ k [ z ] × l suchthat ~ w F ( f , z ) = × m . As all the ( l − × ( l −
1) column reduced minors of F ( f , z ) generate k [ z ], all the 1 × ~ w generate k [ z ] by Lemma 2. Assume that w ∈ k [ z ] is thegreatest common divisor of w , . . . , w l , then ~ w / w is a ZLP vector. Using Quillen-Suslin theorem,we can construct a unimodular matrix U ∈ k [ z ] l × l such that ~ w / w is its first row. This impliesthat there are D ∈ k [ z ] l × l and F ∈ k [ z ] l × m such that UF = DF , where D = diag( h , , . . . , F w.r.t. h , i.e., F = G F , where G = U − D anddet( G ) = h . Let F ∈ M , and a , . . . , a β ∈ k [ z ] be all the l × l minors of F . Since h | d l ( F ), there are e , . . . , e β ∈ k [ z ] such that a i = he i , i = , . . . , β . Lin et al. (2001) proved that F has a matrixfactorization w.r.t. h if h h , e , . . . , e β i = k [ z ]. The main idea is as follows. h h , e , . . . , e β i = k [ z ]implies that rank( F ( f , z )) = l − z ∈ k n − , then we can construct a ZLP vector ~ w ∈ k [ z ] × l such that ~ w F ( f , z ) = × m . Obviously, in this situation, we have h ∤ d l − ( F ). So, thecondition h h , e , . . . , e β i = k [ z ] is a special case of Theorem 22.When d l ( F ) = h , Lin et al. (2005) proved that F has an MLP matrix factorization w.r.t. h ifand only if all the ( l − × ( l −
1) column reduced minors of F ( f , z ) generate k [ z ]. In fact, d l ( F ) = h implies that h ∤ d l − ( F ). Hence, the main result of Lin et al. (2005) is also a specialcase of Theorem 22.Let c , . . . , c η ∈ k [ z ] be all the ( l − × ( l −
1) minors of F . Liu et al. (2011) proved thatrank( F ( f , z )) = l − z ∈ k n − if and only if h h , c , . . . , c η i = k [ z ]. Then, F has amatrix factorization w.r.t. h if h h , c , . . . , c η i = k [ z ]. Although Liu et al. (2011) generalized themain result of Lin et al. (2001), h h , c , . . . , c η i = k [ z ] is still a special case of Theorem 22.Let b , . . . , b η ∈ k [ z ] be all the ( l − × ( l −
1) reduced minors of F . Lu et al. (2020a) provedthat F has a matrix factorization w.r.t. h if h ∤ d l − ( F ) and h h , b , . . . , b η i = k [ z ]. We explain thedi ff erence between h h , c , . . . , c η i = k [ z ] and h h , b , . . . , b η i = k [ z ]. h h , c , . . . , c η i = k [ z ] impliesthat all the ( l − × ( l −
1) minors of F ( f , z ) generate k [ z ], and h h , b , . . . , b η i = k [ z ] impliesthat all the ( l − × ( l −
1) reduced minors of F ( f , z ) generate k [ z ]. Therefore, the main resultof Lu et al. (2020a) is a generalization of that of Liu et al. (2011). Under the assumption that h ∤ d l − ( F ), there is no doubt that h h , b , . . . , b η i = k [ z ] is a special case of Theorem 22.Assume that h ∤ d l − ( F ) and h h , b , . . . , b η i = k [ z ], then there exists a ZLP vector ~ w ∈ k [ z ] × l such that ~ w F ( f , z ) = × m . In Theorem 19, Fitt l − ( W ) = Fitt l − ( W ) = h d i implies thatall the 1 × H generate k [ z ]. Then we can obtain a ZLP vector ~ w ∈ k [ z ] × l by factorizing H . Although the conditions in Lu et al. (2020a) and Theorem 19 all imply that wecan construct a ZLP vector ~ w ∈ k [ z ] × l , h h , b , . . . , b η i = k [ z ] cannot deduce Fitt l − ( W ) = h d i . Itfollows that Theorem 19 is not a generalization of the main result in Lu et al. (2020a). However,Example 12 shows that Theorem 19 can solve some problems that the main result in Lu et al.(2020a) cannot solve.Assume that H ∈ k [ z ] s × l is composed of a system of generators of the syzygy module of F ( f , z ). Then, Syz( F ( f , z )) = Im( H ). Fitt l − ( W ) = h d i in Theorem 19 implies that all the 1 × H generate k [ z ]. According to Lemma 2, all the ( l − × ( l −
1) columnreduced minors of F ( f , z ) generate k [ z ]. Thus, Theorem 19 can deduce Theorem 22. However,9heorem 22 only imply that all the 1 × H generate k [ z ]. It followsthat Theorem 19 is not equivalent to Theorem 22. Therefore, Theorem 19 is a special case ofTheorem 22.Based on Lemma 17, Liu and Wang (2013) proposed a criterion for the existence of a matrixfactorization of F w.r.t. h . Lemma 23.
Let F ∈ k [ z ] l × m be a full row rank matrix, and h ∈ k [ z ] be a divisor of d l ( F ) .a , . . . , a β ∈ k [ z ] and c , . . . , c η ∈ k [ z ] be all the l × l minors and ( l − × ( l − minors of F , re-spectively. There are e , . . . , e β ∈ k [ z ] such that a i = h e i , i = , . . . , β . If h , e , . . . , e β , c , . . . , c η generate k [ z ] , then F has a matrix factorization w.r.t. h . In Lemma 23, F does not have to belong to M and h does not have to be of the form z − f ( z ). Obviously, the main results of Lin et al. (2001) and Liu et al. (2011) are special casesof Lemma 23. When h = z − f ( z ), however, we find that h h , e , . . . , e β , c , . . . , c η i = k [ z ] isequivalent to h h , c , . . . , c η i = k [ z ]. That is, Lemma 23 is the same as the main result of Liu et al.(2011) for the case of h = z − f ( z ). Before proving this conclusion, we first introduce a lemmawhich proposed by Lin et al. (2001). Lemma 24.
Let F ∈ k [ z ] l × m be a univariate polynomial matrix with full row rank, and d ∈ k [ z ] be the greatest common divisor of all the l × l minors of F . If z ∈ k is a simple zero of d, i.e.,z − z is a divisor of d, but ( z − z ) is not a divisor of d, then rank( F ( z )) = l − . Now, we can assert that the following conclusion is correct.
Proposition 25.
Let F ∈ M , a , . . . , a β ∈ k [ z ] and c , . . . , c η ∈ k [ z ] be all the l × l minors and ( l − × ( l − minors of F , respectively. There are e , . . . , e β ∈ k [ z ] such that a i = he i , i = , . . . , β .Then, h h , e , . . . , e β , c , . . . , c η i = k [ z ] if and only if h h , c , . . . , c η i = k [ z ] .Proof. Su ffi ciency is obvious, we next prove the necessity.Assume that h h , e , . . . , e β , c , . . . , c η i = k [ z ]. If h h , c , . . . , c η i , k [ z ], then there exists a point ~ε = ( ε , . . . , ε n ) ∈ k n such that ε = f ( ε , . . . , ε n ) and c i ( ~ε ) = , i = , . . . , η. Then, rank( F ( ~ε )) < l −
1. Let ˜ F = F ( z , ε , . . . , ε n ) be a univariate polynomial matrix with entriesin k [ z ], and ˜ a , . . . , ˜ a β ∈ k [ z ] be all the l × l minors of ˜ F . Obviously, we have˜ a j = a j ( z , ε , . . . , ε n ) = ( z − ε ) · e j ( z , ε , . . . , ε n ) , j = , . . . , β. Assume that q ∈ k [ z ] is the greatest common divisor of e ( z , ε , . . . , ε n ) , . . . , e β ( z , ε , . . . , ε n ),then d l ( ˜ F ) = ( z − ε ) · q . It follows from h h , e , . . . , e β , c , . . . , c η i = k [ z ] that ~ε is not a commonzero of the system { e = , . . . , e β = } . Thus, ε is not a zero of p . This implies that ε is asimple zero of d l ( ˜ F ). According to Lemma 25, we have rank( ˜ F ( ε )) = l −
1, which leads to acontradiction. Therefore, h h , c , . . . , c η i = k [ z ].
4. Equivalence for polynomial matrices
In this section, we first put forward a necessary and su ffi cient condition to solve Problem 16,and then use an example to illustrate the e ff ectiveness of the matrix equivalence theorem.We introduce a lemma, which is called the Binet-Cauchy formula (Strang, 1980).10 emma 26. Let F = G F , where G ∈ k [ z ] l × l and F ∈ k [ z ] l × m . Then an i × i minor of F is det (cid:16) F (cid:16) r ··· r i j ··· j i (cid:17)(cid:17) = X ≤ s < ··· < s i ≤ l det (cid:16) G (cid:16) r ··· r i s ··· s i (cid:17)(cid:17) · det (cid:16) F (cid:16) s ··· s i j ··· j i (cid:17)(cid:17) . In Lemma 26, F (cid:16) r ··· r i j ··· j i (cid:17) denotes an i × i submatrix consisting of the r , . . . , r i rows and j , . . . , j i columns of F . Based on this lemma, we can obtain the following two results. Lemma 27.
Let F ∈ k [ z ] l × m be of full row rank with F = G F , where G ∈ k [ z ] l × l and F ∈ k [ z ] l × m . Then d i ( F ) | d i ( F ) and d i ( G ) | d i ( F ) for each i ∈ { , . . . , l } .Proof. We only prove d i ( F ) | d i ( F ), since the proof of d i ( G ) | d i ( F ) follows in a similar manner.For any given i ∈ { , . . . , l } , let a i , , . . . , a i , t i and ¯ a i , , . . . , ¯ a i , t i be all the i × i minors of F and F respectively, where t i = (cid:16) li (cid:17)(cid:16) mi (cid:17) . For each a i , j , it is a k [ z ]-linear combination of ¯ a i , , . . . , ¯ a i , t i byusing Lemma 26, where j = , . . . , t i . Since d i ( F ) is the greatest common divisor of ¯ a i , , . . . , ¯ a i , t i ,for each j we have d i ( F ) | a i , j . Then, d i ( F ) | d i ( F ). Lemma 28.
Let F , F ∈ k [ z ] l × m be of full row rank. If F and F are equivalent, then d i ( F ) = d i ( F ) for each i ∈ { , . . . , l } .Proof. Since F and F are equivalent, then there exist two unimodular matrices U ∈ k [ z ] l × l and V ∈ k [ z ] m × m such that F = UF V . For each i ∈ { , . . . , l } , it follows from Lemma 27that d i ( F ) | d i ( UF ) | d i ( F ). Furthermore, we have F = U − F V − since U and V are twounimodular matrices. Similarly, we obtain d i ( F ) | d i ( U − F ) | d i ( F ). Therefore, d i ( F ) = d i ( F )up to multiplication by a nonzero constant. Lemma 29 (Lu et al. (2017)) . Let F ∈ k [ z ] l × m with rank l − r. If all the ( l − r ) × ( l − r ) minors of F generate k [ z ] , then there exists a ZLP matrix H ∈ k [ z ] r × l such that HF = r × m . Combining Lemma 29 and the Quillen-Suslin theorem, we can now solve Problem 16.
Theorem 30.
Let F ∈ k [ z ] l × l with det( F ) = h r , where h = z − f ( z ) and ≤ r ≤ l. Then F and diag( h , . . . , h | {z } r , , . . . , | {z } l − r ) are equivalent if and only if h | d l − r + ( F ) and the ideal generated by h andall the ( l − r ) × ( l − r ) minors of F is k [ z ] .Proof. For convenience, let D = diag( h , . . . , h , , . . . ,
1) and ¯ F = F ( f , z ). Let a , . . . , a β be allthe ( l − r ) × ( l − r ) minors of F . It is obvious that a ( f , z ) , . . . , a β ( f , z ) are all the ( l − r ) × ( l − r )minors of ¯ F .Su ffi ciency. It follows from h | d l − r + ( F ) that rank( ¯ F ) ≤ l − r . Assume that there exists a point( ε , . . . , ε n ) ∈ k × ( n − such that a i ( f ( ε , . . . , ε n ) , ε , . . . , ε n ) = , i = , . . . , β. (3)Let ε = f ( ε , . . . , ε n ), then Equation (3) implies that ( ε , ε , . . . , ε n ) ∈ k × n is a common zeroof the polynomial system { h = , a = , . . . , a β = } . This contradicts the fact that h and all the( l − r ) × ( l − r ) minors of F generate k [ z ]. Then, all the ( l − r ) × ( l − r ) minors of ¯ F generate k [ z ].According to Lemma 29, there exists a ZLP matrix H ∈ k [ z ] r × l such that H ¯ F = r × l . Based onthe Quillen-Suslin theorem, we can construct a unimodular matrix U ∈ k [ z ] l × l such that H is itsfirst r rows. Then, there is a polynomial matrix V ∈ k [ z ] l × l such that UF = DV . Since det( F ) = h r U is a unimodular matrix, we have F = U − DV and V is a unimodular matrix. Therefore, F and D are equivalent.Necessity. If F and D are equivalent, then there exist two unimodular matrices U ∈ k [ z ] l × l and V ∈ k [ z ] l × l such that F = UDV . It follows from Lemma 28 that d l − r + ( F ) = d l − r + ( D ) = h . If h h , a , . . . , a β i , k [ z ], then there exists a point ~ε ∈ k × n such that h ( ~ε ) = F ( ~ε )) < l − r .Obviously, rank( D ( ~ε )) = l − r and rank( U − ( ~ε )) = rank( V − ( ~ε )) = l . Since D = U − FV − , wehave rank( D ( ~ε )) ≤ min { rank( U − ( ~ε )) , rank( F ( ~ε )) , rank( V − ( ~ε )) } , which leads to a contradiction. Therefore, h h , a , . . . , a β i = k [ z ] and the proof is completed. Remark 31.
When r = l in Theorem 30, we just need to check whether h is a divisor of d ( F ) . Now, we use Example 15 to illustrate a constructive method which follows the proof pro-cess of the su ffi ciency of Theorem 30 and explain how to obtain the two unimodular matricesassociated with equivalent matrices. Example 32.
Let F be the same polynomial matrix as in Example 15. It is easy to compute that det( F ) = ( z − z ) and d ( F ) = z − z . Let h = z − z , it is obvious that h | d ( F ) . The reducedGr¨obner basis of the ideal generated by h and all the × minors of F w.r.t. ≺ z is { } . Then, F is equivalent to diag( h , h , .Note that F ( z , z , z ) = ( z + z − z ( z −
1) 0 z + z
00 0 0 , rank( F ( z , z , z )) = . Let W = Im( F ( z , z , z )) . We compute a system of generators of thesyzygy module of W, and obtain H = " − z + z − z − z − − z + z + such that H · F ( z , z , z ) = × . It is easy to check that H is a ZLP matrix. Then, a unimodularmatrix U ∈ k [ z ] × can be constructed such that H is its the first rows by using the Maplepackage QUILLENSUSLIN, where U = − z + z − z − z − − z + z + − z − z . Now we can extract h from the first rows of UF , and get F = U − · diag( h , h , · V = z z − z z + h h
00 0 1 z + z z + z .
5. Generalizations
We construct the following two sets of polynomial matrices: M = { F ∈ M : h ∤ d l − ( F ) } and M = { F ∈ M : h | d l − ( F ) } . F ∈ M . Assume that h = z − f ( z ) is given, then F ∈ M or F ∈ M . If F ∈ M , we canuse Theorem 22 to judge whether F has a matrix factorization w.r.t. h . If F ∈ M , we need topropose some criteria to factorize F .Since d ( F ) | d ( F ) | · · · | d l − ( F ) | d l ( F ), there exists a unique integer r with 1 ≤ r ≤ l suchthat h | d l − r + ( F ) but h ∤ d l − r ( F ). Based on this fact, we subdivide M into the following sets: M , r = { F ∈ M : h | d l − r + ( F ) but h ∤ d l − r ( F ) } , r = , . . . , l . Lemma 33.
Let F ∈ M . Then rank( F ( f , z )) = l − r with ≤ r ≤ l if and only if F ∈ M , r . The proof of Lemma 33 is basically the same as that of Lemma 20, so it is omitted here.Inspired by Theorem 22 and Theorem 30, we propose the following result for the existence of amatrix factorization of F ∈ M , r w.r.t. h r , where 2 ≤ r < l . Theorem 34.
Let F ∈ M , r with ≤ r < l, then the following are equivalent: there are G ∈ k [ z ] l × l and F ∈ k [ z ] l × m such that F = G F , and G is equivalent to diag( h , . . . , h | {z } r , , . . . , | {z } l − r ) ; all the ( l − r ) × ( l − r ) column reduced minors of F ( f , z ) generate k [ z ] .Proof. →
2. Since G and diag( h , . . . , h , , . . . ,
1) are equivalent, we have h | d l − r + ( G ) and h h , g , . . . , g η i = k [ z ] by Theorem 30, where g , . . . , g η are all the ( l − r ) × ( l − r ) minors of G .This implies that all the ( l − r ) × ( l − r ) minors of G ( f , z ) generate k [ z ]. According to Lemma29, we can construct a ZLP matrix W ∈ k [ z ] r × l such that WG ( f , z ) = r × l . It follows from F = G F that WF ( f , z ) = r × m . Since W is a ZLP matrix, all the ( l − r ) × ( l − r ) column reducedminors of F ( f , z ) generate k [ z ].2 →
1. From Lemma 33, there exists a full row rank matrix H ∈ k [ z ] r × l such that HF ( f , z ) = r × m . Since all the ( l − r ) × ( l − r ) column reduced minors of F ( f , z ) generate k [ z ], allthe r × r reduced minors of H generate k [ z ] by Lemma 2. Using Lemma 17, H has a ZLP matrixfactorization: H = H H , where H ∈ k [ z ] r × r , and H ∈ k [ z ] r × l is a ZLP matrix. As H is a fullcolumn rank matrix, it follows from HF ( f , z ) = r × m that H F ( f , z ) = r × m . Using Quillen-Suslin theorem, we can construct a unimodular matrix U ∈ k [ z ] l × l such that H is its first r rows.This implies that there is F ∈ k [ z ] l × m such that UF = DF , where D = diag( h , . . . , h , , . . . , D ) = h r . Therefore, we obtain a matrix factorization of F w.r.t. h r , i.e., F = G F with G = U − D . Obviously, G is equivalent to D . Remark 35.
In Theorem 34, the matrix factorization F = G F must satisfies that G is equiva-lent to diag( h , . . . , h , , . . . , . Since there exist many polynomial matrices such that their matrixfactorizations do not satisfy this requirement, the condition “all the ( l − r ) × ( l − r ) column re-duced minors of F ( f , z ) generate k [ z ] ” is only a su ffi cient condition for the existence of a matrixfactorization of F ∈ M , r w.r.t. h r , where ≤ r < l. Theorem 36.
Let F ∈ M , l , then h is a common divisor of all entries in F . We can extract h fromeach row of F and obtain a matrix factorization of F w.r.t. h l . Let k [¯ z i ] = k [ z , . . . , z i − , z i + , . . . , z n ] and h i = z i − f (¯ z i ), where f (¯ z i ) ∈ k [¯ z i ] and 1 ≤ i ≤ n .We construct the following sets of polynomial matrices: M ( i , r ) = { F ∈ k [ z ] l × m : h i | d l − r + ( F ) but h i ∤ d l − r ( F ) } , r = , . . . , l . Then, we can get the following corollary. 13 orollary 37.
Let F ∈ M ( i , r ) , where ≤ i ≤ n and ≤ r ≤ l. If all the ( l − r ) × ( l − r ) column reduced minors of F ( z , . . . , z i − , f , z i + , . . . , z n ) generate k [¯ z i ] , then F admits a matrixfactorization w.r.t. h ri .
6. Factorization algorithm and its uniqueness
In this section, we first propose an algorithm to factorize F ∈ M w.r.t. h r , where 1 ≤ r ≤ l .And then, we study the uniqueness of matrix factorizations by the algorithm. According to Theorem 22, Theorem 34 and Theorem 36, we construct an algorithm to fac-torize polynomial matrices in M . Algorithm 1: factorization algorithm
Input : F ∈ M , h = z − f ( z ) and a monomial order ≺ z in k [ z ]. Output: a matrix factorization of F w.r.t. h r , where 1 ≤ r ≤ l . begin compute the rank l − r of F ( f , z ); if r = l then extract h from each row of F and obtain F , i.e., F = diag( h , . . . , h ) · F ; return diag( h , . . . , h ) and F . compute a reduced Gr¨obner basis G of all the ( l − r ) × ( l − r ) column reduced minorsof F ( f , z ) w.r.t. ≺ z ; if G , { } then if r = then return F has no matrix factorizations w.r.t. h . else return unable to judge. compute a ZLP matrix H ∈ k [ z ] r × l such that HF ( f , z ) = r × m ; construct a unimodular matrix U ∈ k [ z ] l × l such that H is its first r rows; compute F ∈ k [ z ] l × m such that UF = diag( h , . . . , h , , . . . , · F ; return U − · diag( h , . . . , h , , . . . ,
1) and F . Theorem 38.
Algorithm 1 works correctly.Proof.
The proof follows directly from Theorem 22, Theorem 34 and Theorem 36.Before proceeding further, let us remark on Algorithm 1. • It follows from G , { } in Step 7 that all the ( l − r ) × ( l − r ) column reduced minors of F ( f , z ) do not generate k [ z ]. • Under the assumption that G , { } and r =
1, the algorithm in Step 9 returns that “ F has nomatrix factorizations w.r.t. h ” by Theorem 22. When G , { } and 1 < r < l , the algorithmin Step 11 returns that “unable to judge” by Remark 35.14 We explain how to calculate a ZLP matrix H in Step 12. We first compute a Gr¨obnerbasis G ∗ of the syzygy module of F ( f , z ). As rank( F ( f , z )) = l − r , we can select r k [ z ]-linearly independent vectors from G ∗ and form H ∈ k [ z ] r × l with full row rank. Accordingto Lemma 2, all the r × r reduced minors of H generate k [ z ]. Then, H has a ZLP matrixfactorization by Lemma 17. Hence, we second use the Maple package QUILLENSUSLINto compute a ZLP matrix factorization of H and obtain a ZLP matrix H . • In Step 13 we use QUILLENSUSLIN again to construct a unimodular matrix. SinceQUILLENSUSLIN is a Maple package, we implement the factorization algorithm onMaple. Codes and examples are available on the website: . Liu and Wang (2015) studied the uniqueness problem of polynomial matrix factorizations.They pointed out that for a non-regular factor h of F ∈ k [ z ] l × m , under the condition that thereexists a matrix factorization F = G F with det( G ) = h , Im( F ) is not uniquely determined. Inother words, when F = G F = G F with det( G ) = det( G ) = h , Im( F ) and Im( F ) mightnot be the same.Let F ∈ M , h = z − f ( z ) and ≺ z are given. We use Algorithm 1 to factorize F w.r.t. h r ,where 1 ≤ r ≤ l . Assume that all the ( l − r ) × ( l − r ) column reduced minors of F ( f , z ) generate k [ z ], then we need to compute a ZLP matrix and construct a unimodular matrix. Due to thedi ff erent choices of a ZLP matrix and a unimodular matrix, we will get di ff erent matrix factor-izations of F w.r.t. h r . Hence, in the following we study the uniqueness of matrix factorizationsby Algorithm 1. Theorem 39.
Let F ∈ M satisfy F = U − DF = U − DF , where U , U are two unimodularmatrices in k [ z ] l × l , and D = diag( h , . . . , h | {z } r , , . . . , | {z } l − r ) . Then, Im( F ) = Im( F ) .Proof. Let F = h ~ u T1 , . . . , ~ u T l i T and F = h ~ v T1 , . . . ,~ v T l i T , where ~ u , . . . , ~ u l ,~ v , . . . ,~ v l ∈ k [ z ] × m . So,Im( F ) = h ~ u , . . . , ~ u l i and Im( F ) = h ~ v , . . . ,~ v l i .Let F = U F and F = U F . Then F = DF and F = DF . It follows that F = h h ~ u T1 , . . . , h ~ u T r , ~ u T r + , . . . , ~ u T l i T and F = h h ~ v T1 , . . . , h ~ v T r ,~ v T r + , . . . ,~ v T l i T . Since U and U are two unimodular matrices in k [ z ] l × l , we have F = U U − F . This implies that there existpolynomials a i , . . . , a il ∈ k [ z ] such that h ~ u i = h · ( r X j = a i j ~ v j ) + l X j = r + a i j ~ v j , where i = , . . . , r . Then, for each i setting z of the above equation to f ( z ), we have a i ( r + ~ v r + ( f , z ) + · · · + a il ~ v l ( f , z ) = ~ . As rank( F ( f , z )) = l − r and rank( F ( f , z )) = rank( F ( f , z )), we have that ~ v r + ( f , z ) , . . . ,~ v l ( f , z ) are k [ z ]-linearly independent. This implies that a i ( r + = · · · = a il =
0. Hence, ~ u i = a i ~ v + · · · + a ir ~ v r , i = , . . . , r . Obviously, ~ u j is a k [ z ]-linear combination of ~ v , . . . ,~ v l , where j = r + ,. . . , l . As a consequence, h ~ u , . . . , ~ u l i ⊂ h ~ v , . . . ,~ v l i . We can use the same method to prove that h ~ v , . . . ,~ v l i ⊂ h ~ u , . . . , ~ u l i .Therefore, we have Im( F ) = Im( F ).Based on Theorem 39, we can now derive the conclusion: the output F of Algorithm 1 isunique, i.e., Im( F ) is uniquely determined.
7. Examples
We use two examples to illustrate the calculation process of Algorithm 1. We first return toExample 12.
Example 40.
Let F = " − z z + z z + z z − z z + z z z − z − z z + z z z z − z z z − z z + z − z + z z z be a polynomial matrix in C [ z , z , z ] × , where z > z > z and C is the complex field.It is easy to compute that d ( F ) = z ( z − z ) and d ( F ) = . Let F , h = z − z and ≺ z , z bethe inputs of Algorithm 1, where ≺ z , z is the degree reverse lexicographic order.Note that F ( z , z , z ) = " − z z + z z − z + z z z − z z + z − z + z z , rank( F ( z , z , z )) = and r = . All the × column reduced minors of F ( z , z , z ) are z , .Since the reduced Gr¨obner basis of h z , i w.r.t. ≺ z , z is { } , F has a matrix factorization w.r.t. h.Let W = Im( F ( z , z , z )) . Then we compute a reduced Gr¨obner basis of the syzygy moduleof W, and obtain H = h − z i . It is easy to check that H is a ZLP matrix. H can be extended as the first row of a unimodularmatrix U = " − z by using the package QUILLENSUSLIN. We extract h from the first row of UF , and get UF = DF = " z − z
00 1 z z − z z − z z z − z z + z − z + z z z . Then, F has a matrix factorization w.r.t. h: F = G F = ( U − D ) F = " z − z z z z − z z − z z z − z z + z − z + z z z , where det( G ) = det( U − D ) = h.At this moment, d ( F ) = z . We reuse Algorithm 1 to judge whether F has a matrix factor-ization w.r.t. z . Note that F ( z , , z ) = " z z z z z z , F ( z , , z )) = and r = . All the × column reduced minors of F ( z , , z ) are z , z ,and the reduced Gr¨obner basis G of h z , z i is { z , z } . Since G , { } and r = , F has no matrixfactorizations w.r.t. z . Remark 41.
In Example 40, we can first judge whether F has a matrix factorization w.r.t. z .Note that F ( z , , z ) = " z z ( z − z ) z ( z − z ) 0 0 z z z , rank( F ( z , , z )) = and r = . All the × column reduced minors of F ( z , , z ) are z ( z − z ) , z , and do not generate k [ z , z ] . This implies that F has no matrix factorizations w.r.t. z .According to the above calculations, we have the following conclusion: F has a matrix fac-torization w.r.t. z − z , but does not have a matrix factorization w.r.t. z . Example 42.
Let F = z − z z z z + z + z + z − z z − z z z − z − z z + z z z − z z + z z − z z + z − z be a polynomial matrix in C [ z , z , z ] × , where z > z > z and C is the complex field.It is easy to compute that d ( F ) = − z ( z − z ) ( z z + z z + z ) , d ( F ) = z − z andd ( F ) = . Let F , h = z − z and ≺ z , z be the inputs of Algorithm 1, where ≺ z , z is the degreereverse lexicographic order.Note that F ( z , z , z ) = z + z )( z + − z ( z + z + z − z , rank( F ( z , z , z )) = and r = . Obviously, all the × column reduced minors of F ( z , z , z ) are z + , . Since the reduced Gr¨obner basis of h z + , i w.r.t. ≺ z , z is { } , F has a matrixfactorization w.r.t. h .Let W = Im( F ( z , z , z )) . Then we compute a reduced Gr¨obner basis of the syzygy moduleof W, and obtain H = " − z −
10 1 0 . It is easy to check that the reduced Gr¨obner basis of all the × minors of H w.r.t. ≺ z , z is G = { } . Then, H is a ZLP matrix. We use the package QUILLENSUSLIN to construct aunimodular matrix U = − z −
10 1 00 0 1 such that H is the first rows of U . We extract h from the first rows of UF , and get UF = DF = z − z z − z
00 0 1 z z − z z + z z + z − z . Then, we obtain a matrix factorization of F w.r.t. h : = G F = ( U − D ) F = z − z z + z − z
00 0 1 z z − z z + z z + z − z , where det( G ) = det( U − D ) = h .At this moment, d ( F ) = − z ( z z + z z + z ) . We reuse Algorithm 1 to judge whether F has a matrix factorization w.r.t. z . Similarly, we obtain F = G F = z z − z z + z z + z − z , where det( G ) = z .Therefore, we obtain a matrix factorization of F w.r.t. z ( z − z ) , i.e., F = GF = ( G G ) F = z ( z − z ) 0 z + z − z
00 0 1 z − z z + z z + z − z , where det( G ) = z ( z − z ) . Remark 43.
In Example 42, we can first judge whether F has a matrix factorization w.r.t. z .Note that F (0 , z , z ) = z + z )( z + − z ( z + − z z z − z z + z − z , rank( F (0 , z , z )) = and r = . All the × column reduced minors of F (0 , z , z ) are z + , ,and generate k [ z , z ] . This implies that F has a matrix factorization w.r.t. z .According to the above calculations, we have the following conclusion: F has a matrix fac-torization w.r.t. z , z − z , z ( z − z ) , ( z − z ) and z ( z − z ) , respectively.
8. Concluding remarks
In this paper, we point out two directions of research in which multivariate polynomial ma-trices have been explored. The first is concerned with the factorization problem for a class ofmultivariate polynomial matrices, and the second direction is devoted to the investigation of theequivalence problem of a square polynomial matrix and a diagonal matrix.The main contributions of this paper include: 1) some new factorization criteria are given tofactorize F ∈ M w.r.t. h r , and the relationships among all existed factorization criteria have beenstudied; 2) a necessary and su ffi cient condition is proposed to judge whether a square polynomialmatrix with the determinant being h r is equivalent to the diagonal matrix diag( h , . . . , h , , . . . , ff ectiveness of the algorithm.A su ffi cient condition is obtained for the existence of a matrix factorization of F w.r.t. h r (1 < r < l ). At this moment, how to establish a necessary and su ffi cient condition for F admitting amatrix factorization w.r.t. h r is the question that remains for further investigation.18 cknowledgments This research was supported by the CAS Key Project QYZDJ-SSW-SYS022.
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