Factorizations for a Class of Multivariate Polynomial Matrices
aa r X i v : . [ c s . S C ] M a y Factorizations for a Class of Multivariate Polynomial Matrices
Dong Lu a,b , Dingkang Wang a,b , Fanghui Xiao a,b a KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China b School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
Abstract
Following the works by Lin et al. (Circuits Syst. Signal Process. 20(6): 601-618, 2001) and Liuet al. (Circuits Syst. Signal Process. 30(3): 553-566, 2011), we investigate how to factorize aclass of multivariate polynomial matrices. The main theorem in this paper shows that an l × m polynomial matrix admits a factorization with respect to a polynomial if the polynomial and allthe ( l − × ( l −
1) reduced minors of the matrix generate the unit ideal. This result is a furthergeneralization of previous works, and based on this, we give an algorithm which can be used tofactorize more polonomial matrices. In addition, an illustrate example is given to show that ourmain theorem is non-trivial and valuable.
Keywords:
Multivariate polynomial matrices, Matrix factorizations, Reduced minors, ReducedGr¨obner basis
1. Introduction
The study of factorizations for multivariate polynomial matrices began with the developmentof multidimensional system theory in the late 1970s (Youla and Gnavi, 1979), and the problem ofmatrix factorizations was considered to be one of the basic problems of this subject. Since then,great progress has been made on multivariate polynomial matrix factorizations.Bose (1982) introduced some basic concepts of multivariate polynomial matrices and the prob-lem of matrix factorizations. After that, Bose et al. (2003) presented factorization algorithms ofbivariate polynomial matrices, and introduced the latest research trends of matrix factorizationswith three or more variables. The factorization problem for bivariate polynomial matrices has beencompletely solved in (Guiver and Bose, 1982; Liu and Wang, 2013; Morf et al., 1977), but for thecases of more than two variables is still open.Charoenlarpnopparut and Bose (1999) used Gr¨obner bases of modules to compute zero primematrix factorizations of multivariate polynomial matrices. For some polynomial matrices withspecial properties, Lin (1999a, 2001) proposed some methods to compute zero prime matrix fac-torizations of matrices. Meanwhile, Lin and Bose (2001) presented the Lin-Bose’s conjecture: amatrix admits a zero prime matrix factorization if its all maximal reduced minors generate the unitideal. This conjecture was proved in (Liu et al., 2014; Pommaret, 2001; Wang and Feng, 2004),so the problem of zero prime matrix factorizations have been completely solved. Subsequently,Wang and Kwong (2005) put forward an algorithm based on module theory to solve the problem
Email addresses: [email protected] (Dong Lu), [email protected] (Dingkang Wang), [email protected] (Fanghui Xiao)
Preprint submitted to Multidimensional Systems and Signal Processing May 29, 2019 f minor prime matrix factorizations. Guan et al. (2018, 2019) studied the problem of factor primematrix factorizations under the condition that matrices are not of full rank, and they generalizedthe main results in (Wang and Kwong, 2005). So far, some achievements in (Liu and Wang, 2010,2015; Wang, 2007) have been made on the problem of factor prime matrix factorizations. Althoughthe problems of zero prime matrix factorizations and minor prime matrix factorizations have beencompletely solved, the problem of factor prime matrix factorizations remains to be studied.Let k [ z ] and k [ z ] be the ring of polynomials in variables z , z , . . . , z n and z , . . . , z n withcoefficients in an algebraically closed field k , respectively. Let F be an l × m polynomial matrixwith entries in k [ z ] and l ≤ m , d l ( F ) be the greatest common divisor of all the l × l minors of F , and d = z − f ( z ) be a divisor of d l ( F ), where f ( z ) ∈ k [ z ]. Lin et al. (2001) proved that F admits amatrix factorization with respect to d if for each ( z ) ∈ k × ( n − the rank of F ( f ( z ) , z ) is ( l − d and all the ( l − × ( l −
1) minors of F , and showed that F admits a matrix factorization with respect to d if d and all the ( l − × ( l − F generate k [ z ]. They proved that their main theorem is a generalization of the result in(Lin et al., 2001). However, we find that there are still many of multivariate polynomial matricesthat can be factorized with respect to d without satisfying the main theorem in (Liu et al., 2011).This implies that it would be significant to generalize the theorems and algorithms in (Lin et al.,2001; Liu et al., 2011).In this paper, we still study the condition under which F admits a matrix factorization withrespect to d . We focus on the relationship between d and all the ( l − × ( l −
1) reduced minors of F , and prove that F admits a matrix factorization with respect to d if d and all the ( l − × ( l − F generate the unit ideal. Compared with the main theorems in (Lin et al.,2001; Liu et al., 2011), our main theorem has a wider range of applications. Combining our maintheorem and the constructive algorithm in (Lin et al., 2001), we obtain the matrix factorizationalgorithm.This paper is organized as follows. In Section 2, we outline some knowledge about multivariatepolynomial matrix factorizations and propose a problem that we shall consider. Main theorem andsome generalizations are presented in Section 3 to help us summarize which types of polynomialmatrices can be factorized. The matrix factorization algorithm is given in Section 4, and an exampleis given to illustrate the calculation process of the algorithm. Further remarks are provided inSection 5.
2. Preliminaries and Problem
In the following, we denote by k an algebraically closed field, z the n variables z , z , . . . , z n , z the ( n −
1) variables z , . . . , z n , where n ≥
3. Let k [ z ] and k [ z ] be the ring of polynomials invariables z and z with coefficients in k , respectively, k ( z ) be the fraction field of k [ z ], and k [ z ] l × m be the set of l × m matrices with entries in k [ z ]. Without loss of generality, we assume that l ≤ m ,and for convenience we use uppercase bold letters to denote polynomial matrices.Throughout this paper, the argument ( z ) is omitted whenever its omission does not causeconfusion. For any given F ∈ k [ z ] l × m and f ( z ) ∈ k [ z ], F T represents the transposed matrix of F ,and F ( f ( z ) , z ) denotes an l × m polynomial matrix in k [ z ] l × m , which is formed by transforming z in F into f ( z ). If l = m , we denote by det( F ) the determinant of F , and if F is of full rank,we use F − ∈ k ( z ) l × l to stand for the inverse matrix of F . Assume that f , . . . , f s ∈ k [ z ], we use h f , . . . , f s i to denote the ideal generated by f , . . . , f s in k [ z ]. Let f, g ∈ k [ z ], then f | g means2hat f is a divisor of g . In addition, “w.r.t.” and “GCD” stand for “with respect to” and “greatestcommon divisor”, respectively. We first introduce two basic concepts in matrix theory.
Definition 2.1.
Let F ∈ k [ z ] l × m , and given r positive integers arbitrarily such that ≤ i < · · · < i r ≤ l and ≤ j < · · · < j r ≤ m . Let F (cid:16) i ··· i r j ··· j r (cid:17) denotes an r × r matrix consisting of the i , . . . , i r rows and j , . . . , j r columns of F , then det (cid:16) F (cid:16) i ··· i r j ··· j r (cid:17)(cid:17) is called an r × r minor of F . Definition 2.2.
Let F ∈ k [ z ] l × m , the rank of F is r (1 ≤ r ≤ l ) if there exists a nonzero r × r minor of F , and all the i × i ( i > r ) minors of F vanish identically. For convenience, we denotethe rank of F by rank( F ) . The following lemma is a generalization of Binet-Cauchy formula in (Strang, 2010).
Lemma 2.3.
Let F = G F ∈ k [ z ] l × m , where G ∈ k [ z ] l × l and F ∈ k [ z ] l × m . Then an r × r ( r ≤ l ) minor of F is det (cid:16) F (cid:16) i ··· i r j ··· j r (cid:17)(cid:17) = X ≤ s < ···
For any given F ∈ k [ z ] l × m , let d l ( F ) and d l − ( F ) be the GCD of all the l × l minors and all the ( l − × ( l − minorsof F , respectively; let a , . . . , a η ∈ k [ z ] be all the l × l minors of F , where η = (cid:0) ml (cid:1) , and extracting d l ( F ) from a , . . . , a η yields a i = d l ( F ) b i , i = 1 , . . . , η, then b , . . . , b η are called the l × l reduced minors of F ; let c , . . . , c γ ∈ k [ z ] be all the ( l − × ( l − minors of F , where γ = (cid:0) ll − (cid:1) · (cid:0) ml − (cid:1) , andextracting d l − ( F ) from c , . . . , c γ yields c i = d l − ( F ) h i , i = 1 , . . . , γ, then h , . . . , h γ are called the ( l − × ( l − reduced minors of F . Now, we introduce two important lemmas in matrix theory.
Lemma 2.5 (Lin (1993, 1999b)) . Let F = [ F , F ] ∈ k [ z ] l × ( m + l ) be of full row rank and F =[ F T21 , − F T22 ] T ∈ k [ z ] ( m + l ) × m be of full column rank, where F , F ∈ k [ z ] l × m , F ∈ k [ z ] l × l and F ∈ k [ z ] m × m . If F F = l × m , then det( F ) = 0 if and only if det( F ) = 0 . emma 2.6 (Lin (1988)) . Assume that F − F = F F − , where F , F ∈ k [ z ] l × m , F − ∈ k ( z ) l × l and F − ∈ k ( z ) m × m . Let ¯ p , . . . , ¯ p ξ be all the l × l reduced minors of [ F , F ] , and p , . . . , p ξ be all the m × m reduced minors of [ F T21 , − F T22 ] T , where ξ = (cid:0) m + ll (cid:1) = ξ = (cid:0) m + lm (cid:1) . Then, ¯ p i = ± p i for i = 1 , . . . , ξ , and the sign depends on the index i . The general matrix factorization problem is now formulated as follows.
Definition 2.7.
Let F ∈ k [ z ] l × m and d ∈ k [ z ] be a divisor of d l ( F ) . We say that F admits amatrix factorization w.r.t. d if F can be factorized as F = G F such that G ∈ k [ z ] l × l , F ∈ k [ z ] l × m , and det( G ) = d . Next we recall the concept of zero left prime matrix from multidimensional systems theory.
Definition 2.8.
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. In Definition 2.7 if F is a ZLP matrix, then we say that F admits a ZLP matrix factorization.Let I be an ideal generated by all the l × l minors of F , then we can compute the reduced Gr¨obnerbasis G of I w.r.t. a term order to check I = k [ z ]. That is, if G = { } , then I = k [ z ]. The definitionof reduced Gr¨obner basis and how to compute a reduced Gr¨obner basis of an ideal can be foundin (Buchberger, 1965; Cox et al., 2007).Serre (1955) raised the question whether any finitely generated projective module over a poly-nomial ring is free. This question was solved positively and independently by Quillen (1976) andSuslin (1976), and the result is called Quillen-Suslin theorem. For Quillen-Suslin theorem, thereare two descriptions as follows. Lemma 2.9. If w ∈ k [ z ] × l is a ZLP vector, then the set M ⊂ k [ z ] l × constructed by all solutions q ∈ k [ z ] l × of wq = 0 is free. Lemma 2.10. If w ∈ k [ z ] × l is a ZLP vector, then an unimodular matrix U ∈ k [ z ] l × l can beconstructed such that w is its first row. In Lemma 2.9, M is called the syzygy module of w . Fabia´nska and Quadrat (2006) gave analgorithm to compute free bases of free modules over polynomial rings, and the algorithm wasimplemented in QuillenSuslin package (Fabia´nska and Quadrat, 2007). In Lemma 2.10, U is anunimodular matrix if and only if det( U ) is a nonzero constant in k . There are many methods toconstruct U such that w is its first row, we refer to (Logar and Sturmfels, 1992; Lu et al., 2017;Park, 1995; Youla and Pickel, 1984) for more details. In order to raise the problem we are going to consider, let us first introduce the works in(Lin et al., 2001) and (Liu et al., 2011).
Lemma 2.11 (Lin et al. (2001)) . Let F ∈ k [ z ] l × m , and d = z − f ( z ) be a common divi-sor of a , . . . , a η , i.e., a i = de i with e i ∈ k [ z ] ( i = 1 , . . . , η ) . If h d, e , . . . , e η i = k [ z ] , then rank( F ( f ( z ) , z )) = l − for every ( z ) ∈ k × ( n − and F admits a matrix factorization w.r.t. d . F ( f ( z ) , z )) = l − z ) ∈ k × ( n − if and only if h d, c , . . . , c γ i = k [ z ]. Therefore, they generalized Lemma 2.11 and obtained the following result. Lemma 2.12 (Liu et al. (2011)) . Let F ∈ k [ z ] l × m , and d = z − f ( z ) be a divisor of d l ( F ) . If h d, c , . . . , c γ i = k [ z ] , then rank( F ( f ( z ) , z )) = l − for every ( z ) ∈ k × ( n − and F admits amatrix factorization w.r.t. d . In the following, let d = z − f ( z ) with f ( z ) ∈ k [ z ]. According to Lemma 2.11 and Lemma2.12, we construct two sets of multivariate polynomial matrices: (cid:26) S := { F ∈ k [ z ] l × m : d | d l ( F ) and h d, e , . . . , e η i = k [ z ] } , S := { F ∈ k [ z ] l × m : d | d l ( F ) and h d, c , . . . , c γ i = k [ z ] } . Then, we have S ⊂ S and F ∈ S admits a matrix factorization w.r.t. d . Example 1 in theSection 4 of (Lin et al., 2001) shows that S is not empty, and Example 4.1 in the Section 4 of(Liu et al., 2011) shows that S $ S .Lemma 2.12 tell us that for any given F ∈ S , rank( F ( f ( z ) , z )) = l −
1. This implies thatGCD( d, d l − ( F )) = 1. Otherwise, it follows from d is an irreducible polynomial that GCD( d, d l − ( F ))= d , then c i ( f ( z ) , z ) = 0 ( i = 1 , . . . , γ ) and rank( F ( f ( z ) , z )) < l −
1, which leads to a contra-diction. Now, we construct a new set of multivariate polynomial matrices: S := { F ∈ k [ z ] l × m : d | d l ( F ) and GCD( d, d l − ( F )) = 1 } . Then, ∅ 6 = S $ S ⊂ S . As we know, d l − ( F ) is the GCD of c , . . . , c γ , then we have h d, c , . . . , c γ i ⊆ h d, d l − ( F ) i ⊆ k [ z ] . Therefore, it follows that h d, c , . . . , c γ i 6 = k [ z ] if h d, d l − ( F ) i 6 = k [ z ]. Although GCD( d, d l − ( F )) = 1for F ∈ S , d and d l − ( F ) may have common zeros. Next, we give an example to show that thereexits F ∈ S \ S such that F admits a matrix factorization w.r.t. d . Example 2.13.
Let F = z z − z − z − z z z z + z − z z − z − z − z F [1 , − z z − z z + z + z z + z z z + z z z − z + z + z − z + z + z + 1 , where F [1 ,
3] = − z z + z z + 2 z + z − z − z − z − .It is easy to compute that d ( F ) = ( z − z )( z + z ) and d ( F ) = z + z . Let d = z − z ,then d | d ( F ) and GCD( d, d ( F )) = 1 . Hence, F ∈ S . a = ( z − z )( z + z ) is the × minor of F , and extracting d from a yields e = ( z + z ) . It is easy to check that the reduced Gr¨obner basis of h d, e i w.r.t. the lexicographic order is { z − z , ( z + z ) } , then F / ∈ S .Since the reduced Gr¨obner basis of h d, d ( F ) i w.r.t. the lexicographic order is { z + z , z + z } ,we have h d, c , . . . , c i ⊆ h d, d ( F ) i 6 = k [ z ] . Then, F / ∈ S .However, we can get a matrix factorization of F w.r.t. d : F = d − z −
10 1 00 0 1 z + z − z − z − z z − z z + z + z z + z z z + z z z − z + z + z − z + z + z + 1 . In Example 2.13, we find that the reduced Gr¨obner basis of h d, h , . . . , h i w.r.t. the lexico-graphic order is { } . In spire of it, we consider the following question. Question 2.14.
Let F ∈ S . If h d, h , . . . , h γ i = k [ z ] , does F have a matrix factorization w.r.t. d ? . Main Results Before giving the main theorem, we introduce two important lemmas.
Lemma 3.1 (Lin et al. (2001)) . Let g ∈ k [ z ] and f ( z ) ∈ k [ z ] . If g ( f, z , . . . , z n ) is a zeropolynomial in k [ z ] , then ( z − f ( z )) is a divisor of g . The following lemma is a generalization of Lemma 2 in (Lin et al., 2001).
Lemma 3.2.
Let F ∈ k [ z ] l × m with rank( F ) = l − . If h h , . . . , h γ i = k [ z ] , then there is a ZLP vector w ∈ k [ z ] × l such that wF = × m .Proof. In view of rank( F ) = l −
1, we could assume that the first ( l −
1) row vectors f , . . . , f l − of F are k [ z ]-linearly independent. This implies that f , . . . , f l − and f l are k [ z ]-linearly dependent.Thus wF = × m for some nonzero row vector w = [ w , . . . , w l ] ∈ k [ z ] × l , where w l = 0 andGCD( w , . . . , w l ) = 1. Obviously, w , . . . , w l are all the 1 × w .The next thing is to prove that w , . . . , w l generate k [ z ]. Let F , . . . , F β ∈ k [ z ] l × ( l − be all the l × ( l −
1) submatrices of F , where β = (cid:0) ml − (cid:1) . For each 1 ≤ i ≤ β , let c i , . . . , c il and h i , . . . , h il beall the ( l − × ( l −
1) minors and all the ( l − × ( l −
1) reduced minors of F i respectively, then c ij = d l − ( F i ) · h ij , where 1 ≤ j ≤ l . Let w = [ w , w l ], where w = [ w , . . . , w l − ] ∈ k [ z ] × ( l − .Let F i = [ F T i , − F T i ] T , where F i ∈ k [ z ] ( l − × ( l − and F i ∈ k [ z ] × ( l − . If F i is not of full columnrank, then c ij = 0 and h ij = 0, j = 1 , . . . , l . If F i is of full column rank, then it follows from wF = × m that (cid:2) w , w l (cid:3) (cid:20) F i − F i (cid:21) = × ( l − . (3.1)Since w l = 0, det( F i ) = 0 by Lemma 2.5. From equation (3.1) we have w − l w = F i F − i . (3.2)According to Lemma 2.6, all the 1 × w are equal to all the ( l − × ( l − F i without considering the sign, i.e., w j = h ij for j = 1 , . . . , l . Therefore, all the( l − × ( l −
1) minors of F are as follows: d l − ( F ) · w , . . . , d l − ( F ) · w l , · · · , d l − ( F β ) · w , . . . , d l − ( F β ) · w l . Let ¯ d ∈ k [ z ] be the GCD of d l − ( F ) , . . . , d l − ( F β ), then there exists ¯ d i ∈ k [ z ] such that d l − ( F i ) =¯ d · ¯ d i , where i = 1 , . . . , β . In the following we prove that the polynomials¯ d w , ¯ d w , · · · ¯ d w l , ¯ d w , ¯ d w , · · · ¯ d w l , ... ... . . . ...¯ d β w , ¯ d β w , · · · ¯ d β w l , are all the ( l − × ( l −
1) reduced minors of F . It follows from GCD( w , . . . , w l ) = 1 andGCD( ¯ d , · · · , ¯ d β ) = 1 thatGCD( ¯ d w , . . . , ¯ d w l , · · · , ¯ d β w , . . . , ¯ d β w l )=GCD(GCD( ¯ d w , . . . , ¯ d w l ) , · · · , GCD( ¯ d β w , . . . , ¯ d β w l ))=GCD( ¯ d , · · · , ¯ d β )=1 . d w , . . . , ¯ d w l , · · · , ¯ d β w , . . . , ¯ d β w l are all the ( l − × ( l −
1) reduced minors of F , i.e.,they are equal to h , . . . , h γ . Since h h , . . . , h γ i = k [ z ], w , . . . , w l generate k [ z ].Combining Lemma 3.1 and Lemma 3.2, we can answer Question 2.14. Theorem 3.3.
Let F ∈ S . If h d, h , . . . , h γ i = k [ z ] , then F admits a matrix factorization w.r.t. d .Proof. We divide our proof into three steps.First, let ˆ F = F ( f ( z ) , z ) ∈ k [ z ] l × m , and we prove that rank( ˆ F ) = l −
1. Let ˆ a , . . . , ˆ a η ∈ k [ z ]and ˆ c , . . . , ˆ c γ ∈ k [ z ] be all the l × l minors and all the ( l − × ( l −
1) minors of ˆ F , respectively.Then, ˆ a i = a i ( f ( z ) , z ) and ˆ c j = c j ( f ( z ) , z ), where 1 ≤ i ≤ η and 1 ≤ j ≤ γ . Since F ∈ S , wehave d | d l ( F ) and GCD( d, d l − ( F )) = 1. d | d l ( F ) implies that ˆ a i = a i ( f ( z ) , z ) = 0 ( i = 1 , . . . , η )and rank( ˆ F ) ≤ l −
1. If rank( ˆ F ) < l −
1, then c j ( f ( z ) , z ) = ˆ c j = 0 ( j = 1 , . . . , γ ). It followsfrom Lemma 3.1 that d is a common divisor of c , . . . , c γ , then d | d l − ( F ), which contradictsGCD( d, d l − ( F )) = 1. Therefore, rank( ˆ F ) = l − l − × ( l −
1) reduced minors of ˆ F generate k [ z ]. Let ¯ h ∈ k [ z ]be the GCD of h ( f ( z ) , z ) , . . . , h γ ( f ( z ) , z ), then for each 1 ≤ j ≤ γ there exits ˆ h j ∈ k [ z ]such that h j ( f ( z ) , z ) = ¯ h · ˆ h j , and GCD(ˆ h , . . . , ˆ h γ ) = 1. Let g = d l − ( F ), then it followsfrom ˆ c j = g ( f ( z ) , z ) · h j ( f ( z ) , z ) that d l − ( ˆ F ) = g ( f ( z ) , z ) · ¯ h , and ˆ h , . . . , ˆ h γ are all the( l − × ( l −
1) reduced minors of ˆ F . Assume that h ˆ h , . . . , ˆ h γ i 6 = k [ z ], then there exists a point( α , . . . , α n ) ∈ k × ( n − such that ˆ h j ( α , . . . , α n ) = 0, where j = 1 , . . . , γ . Let α = f ( α , . . . , α n ),then for each j we have h j ( α , α , . . . , α n ) = ¯ h ( α , . . . , α n ) · ˆ h j ( α , . . . , α n ) = 0. This impliesthat ( α , α , . . . , α n ) ∈ k × n is a common zero of d, h , . . . , h γ , which contradicts the fact that h d, h , . . . , h γ i = k [ z ].Finally, we remark that F has a matrix factorization w.r.t. d . Using Lemma 3.2, we get w ˆ F = × m , where w ∈ k [ z ] × l is a ZLP vector. Meanwhile, according to Lemma 2.10, aunimodular matrix U ∈ k [ z ] l × l can be constructed such that w is its first row. Let F = UF ,then the first row of F ( f ( z ) , z ) = U ˆ F is a zero vector. By Lemma 3.1, d is a common divisor ofthe polynomials in the first row of F , thus F = UF = DF = d ¯ f · · · ¯ f m f · · · f m ... . . . ... f l · · · f lm . Consequently, we can now derive the matrix factorization of F w.r.t. d : F = G F , where G = U − D ∈ k [ z ] l × l , F ∈ k [ z ] l × m and det( G ) = d . Remark 3.4.
In Theorem 3.3, we have that rank( ˆ F ) = l − and h ˆ h , . . . , ˆ h γ i = k [ z ] . Hence,Theorem 3.3 is a generalization of Lemma 2.12. According to Theorem 3.3, we construct a set of multivariate polynomial matrices: S := { F ∈ S : h d, h , . . . , h γ i = k [ z ] } . S ⊂ S ⊂ S and F ∈ S admits a matrix factorization w.r.t. d . Example 2.13 in Section2.2 shows that S $ S .Let F ∈ k [ z ] l × m , and d = Q st =1 ( z − f t ( z )) be a divisor of d l ( F ), where f ( z ) , . . . , f s ( z ) ∈ k [ z ]. Liu et al. (2011) proved that if h d , c , . . . , c γ i = k [ z ], then F admits a matrix factorizationw.r.t. d . It would be interesting to know whether Theorem 3.3 can be generalized to the casewith t >
1. Without loss of generality, we consider the case of t = 2. Theorem 3.5.
Let F ∈ k [ z ] l × m and d = ( z − f ( z ))( z − f ( z )) be a divisor of d l ( F ) . If GCD( d , d l − ( F )) = 1 and h d , h , . . . , h γ i = k [ z ] , then F admits a matrix factorization w.r.t. d .Proof. Let d = z − f ( z ) and d = z − f ( z ). Obviously, GCD( d , d l − ( F )) = 1 and h d , h , . . . , h γ i = k [ z ]. By Theorem 3.3, there exist G ∈ k [ z ] l × l and F ∈ k [ z ] l × m such that F = G F , where G = U − D , det( G ) = d , U ∈ k [ z ] l × l is a unimodular matrix and D = diag( d , , . . . , d = z − f ( z ) is a divisor of d l ( F ). Next we provethat F admits a matrix factorization w.r.t. d .We first prove that GCD( d , d l − ( F )) = 1. Otherwise, it follows from d is an irreduciblepolynomial that GCD( d , d l − ( F )) = d . Then d l − ( F ) | d l − ( F ) implies that d | d l − ( F ), whichcontradicts the condition GCD( d, d l − ( F )) = 1. Second, we prove that d and all the ( l − × ( l − F generate the unit ideal k [ z ].Let F i ∈ k [ z ] ( l − × m be a submatrix obtained by removing the i -th row of F , and ¯ c i , . . . , ¯ c iβ be all the ( l − × ( l −
1) minors of F i , where i = 1 , . . . , l . Then, ¯ c , . . . , ¯ c β , . . . , ¯ c l , . . . , ¯ c lβ areall the ( l − × ( l −
1) minors of F . Extracting d l − ( F ) from ¯ c ij yields ¯ c ij = d l − ( F ) · ¯ h ij , then¯ h , . . . , ¯ h β , . . . , ¯ h l , . . . , ¯ h lβ are all the ( l − × ( l −
1) reduced minors of F . Hence, we only needto prove that h d , ¯ h , . . . , ¯ h lβ i = k [ z ].Since D = diag( d , , . . . , l − × ( l −
1) minors of D F are¯ c , . . . , ¯ c β , d ¯ c , . . . , d ¯ c β , . . . , d ¯ c l , . . . , d ¯ c lβ . Obviously, there is at least one integer j ∈ { , . . . , β } such that d ∤ ¯ c j . Otherwise, d | d l − ( D F ).It follows form F = U − D F and the Equation (2.1) in Lemma 2.3 that d l − ( D F ) | d l − ( F ).So d | d l − ( F ), which leads to a contradiction. Since d = z − f ( z ) is an irreducible polynomial,we have GCD(¯ c , . . . , ¯ c β , d ¯ c , . . . , d ¯ c β , . . . , d ¯ c l , . . . , d ¯ c lβ )=GCD(¯ c , . . . , ¯ c β , ¯ c , . . . , ¯ c β , . . . , ¯ c l , . . . , ¯ c lβ ) , Therefore, d l − ( D F ) = d l − ( F ). It follows from U F = D F that d l − ( F ) | d l − ( D F )and d l − ( F ) = d l − ( F ). The Equation (2.1) in Lemma 2.3 implies that each c i is a k [ z ]-linearcombination of ¯ c , . . . , ¯ c lβ , where i = 1 , . . . , γ . Since d l − ( F ) = d l − ( F ), we can obtain thateach h i (1 ≤ i ≤ γ ) is a k [ z ]-linear combination of ¯ h , . . . , ¯ h lβ . By h d , h , . . . , h γ i = k [ z ], h d , h , . . . , h γ i = k [ z ]. If h d , ¯ h , . . . , ¯ h lβ i 6 = k [ z ], then there exits a point ( α , . . . , α n ) ∈ k × n such that ¯ h ij ( α , . . . , α n ) = 0 for each i and j , where α = f ( α , . . . , α n ). This implies that( α , . . . , α n ) is a common zero of d , h , . . . , h γ , which leads to a contradiction.8ccording to Theorem 3.3 again, there exits G ∈ k [ z ] l × l and F ∈ k [ z ] l × m such that F = G F , where G = U − D , det( G ) = d , U ∈ k [ z ] l × l is an unimodular matrix and D =diag( d , , . . . , F w.r.t. d : F = G F , where G = G G ∈ k [ z ] l × l , and det( G ) = d = ( z − f ( z ))( z − f ( z )). Remark 3.6.
In the above theorem, we can factorize F w.r.t. d without checking whether GCD( d , d l − ( F )) = 1 and the ideal generated by d and all the ( l − × ( l − reduced minors of F is k [ z ] , which can help us improve the computational efficiency of matrix factorizations. It is worth noting that if f ( z ) = f ( z ) in Theorem 3.5, we have the following corollary. Corollary 3.7.
Let F ∈ k [ z ] l × m and d = ( z − f ( z )) r be a divisor of d l ( F ) . If GCD( d , d l − ( F )) =1 and h d , h , . . . , h γ i = k [ z ] , then F admits a matrix factorization w.r.t. d . Further, if f ( z ) = f ( z ) in Theorem 3.5, we have another corollary. Corollary 3.8.
Let F ∈ k [ z ] l × m and d = Q st =1 ( z − f t ( z )) q t be a divisor of d l ( F ) . If GCD( d , d l − ( F ))= 1 and h d , h , . . . , h γ i = k [ z ] , then F admits a matrix factorization w.r.t. d . Let f ( i ) ( z ) be a polynomial in k [ z , . . . , z i − , z i +1 , . . . , z n ], where 1 ≤ i ≤ n . Similarly, we canget the following corollaries. Corollary 3.9.
Let F ∈ k [ z ] l × m and d = ( z i − f ( i ) ( z )) r be a divisor of d l ( F ) . If GCD( d , d l − ( F )) =1 and h d , h , . . . , h γ i = k [ z ] , then F admits a matrix factorization w.r.t. d . Corollary 3.10.
Let F ∈ k [ z ] l × m and d = Q ni =1 Q s i t =1 ( z i − f ( i ) t ( z )) q it be a divisor of d l ( F ) . If GCD( d , d l − ( F )) = 1 and h d , h , . . . , h γ i = k [ z ] , then F admits a matrix factorization w.r.t. d .
4. Algorithm and Example
According to Theorem 3.3, we get the following algorithm for computing a matrix factorizationof F ∈ S w.r.t. d . Algorithm 1:
Matrix Factorization Algorithm
Input : F ∈ S . Output: a matrix factorization of F w.r.t. d . begin compute a ZLP vector w ∈ k [ z ] × l such that wF ( f ( z ) , z ) = × m ; construct a unimodular matrix U ∈ k [ z ] l × l such that w is its first row; compute F ∈ k [ z ] l × m such that UF = DF , where D = diag( d, , . . . , return F = G F , where G = U − D and det( G ) = d .In the following, we show how to compute w and U in Algorithm 1. Let ˆ F = F ( f ( z ) , z ) ∈ k [ z ] l × m and Syz l ( ˆ F ) be the left syzygy module of ˆ F , i.e., Syz l ( ˆ F ) = { p ∈ k [ z ] × l | p ˆ F = × m } .9ince rank( ˆ F ) = l −
1, we have rank(Syz l ( ˆ F )) = 1. Then, we compute a reduced Gr¨obner basisof Syz l ( ˆ F ) w.r.t. a term order, and select a nonzero vector from the Gr¨obner basis. Let w =[ w , . . . , w l ] ∈ k [ z ] × l be the nonzero vector, and w ∈ k [ z ] be the GCD of w , . . . , w l , then w = w w .Since w is a ZLP vector, there exists a column vector q ∈ k [ z ] l × such that wq = 1. Thiscalculation problem is equivalent to a lifting homomorphism problem in (Decker and Lossen, 2006)(see Problem 4.1, page 129), and the command “lift” of the computer algebra system Singular in (Decker et al., 2016) can help us compute q . Let Syz r ( w ) = { q ∈ k [ z ] l × | wq = 0 } , thenSyz r ( w ) is a free module with rank(Syz r ( w )) = ( l −
1) by Lemma 2.9. Let q , . . . , q l ∈ k [ z ] l × bea free basis of Syz r ( w ), then V = [ q , q , . . . , q l ] ∈ k [ z ] l × l is a unimodular matrix and U = V − is one that we want by Theorem 4.4 in (Lu et al., 2017).Now, we use an example to illustrate the calculation process of Algorithm 1. We return toExample 2.13, and let F be the same matrix in Example 2.13. Example 4.1.
Let F = z z − z − z − z z z z + z − z z − z − z − z F [1 , − z z − z z + z + z z + z z z + z z z − z + z + z − z + z + z + 1 , where F [1 ,
3] = − z z + z z + 2 z + z − z − z − z − .As already noted in Example 2.13, h d, h , . . . , h i = k [ z , z , z ] implies that F ∈ S . Then, wecan use Algorithm 1 to factorize F w.r.t. d . Step 1:
Let ˆ F = F ( z , z , z ) ∈ k [ z , z ] × , we compute a ZLP vector w ∈ k [ z , z ] × suchthat w ˆ F = × , where ˆ F = − z ( z + 1) − z ( z + 1) − ( z − z + 1)( z + 1)(1 − z )( z + z ) z + z z ( z + z ) z z z − z + 1 . We use
Singular command “syz” to compute a reduced Gr¨obner basis of
Syz l ( ˆ F ) w.r.t. the lexico-graphic order, and obtain w = [1 , , z + 1] . Step 2:
Construct a unimodular matrix U ∈ k [ z , z ] × such that w is its first row. Accordingto the instruction of the construction for unimodular matrix U below Algorithm 1, we divide it intothree small steps.Step 2.1: Using Singular command “lift” to compute q ∈ k [ z , z ] × such that wq = 1 , weget q = [1 , , T .Step 2.2: Using QuillenSuslin package to compute a free basis of Syz r ( w ) , we have q = [0 , , T and q = [ − ( z + 1) , , T .Step 2.3: Let V = [ q , q , q ] , then U = V − = z + 10 1 00 0 1 . Step 3.
Extracting d from the first row of UF , we get UF = DF , where D = diag( d, , and F = z + z − z − z − z z − z z + z + z z + z z z + z z z − z + z + z − z + z + z + 1 . hen, we obtain a matrix factorization of F w.r.t. d : F = GF = d − z −
10 1 00 0 1 z + z − z − z − z z − z z + z + z z + z z z + z z z − z + z + z − z + z + z + 1 , where G = U − D and det( G ) = d = z − z .
5. Conclusions
We have studied the problem of matrix factorizations for multivariate polynomial matricesin S , and the results presented in this paper greatly extend those of (Lin et al., 2001; Liu et al.,2011). The matrix factorizations for an arbitrary multivariate polynomial matrix remains a chal-lenging and an important open problem. Although the new results can only deal with the class ofmultivariate polynomial matrices discussed in S , we hope that the new results will motivate newprogress in this important research topic. Acknowledgements
This research was supported by the CAS Project QYZDJ-SSW-SYS022.
References
Bose, N., 1982.
Applied multidimensional systems theory . Van Nostrand Reinhold, New York.Bose, N., Buchberger, B., Guiver, J., 2003.
Multidimensional systems theory and applications . Dordrecht, The Nether-lands: Kluwer.Buchberger, B., 1965.
Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimen-sionalen Polynomideal . Ph.D. thesis, Universitat Innsbruck, Austria.Charoenlarpnopparut, C., Bose, N., 1999. Multidimensional FIR filter bank design using Gr¨obner bases.
IEEETransactions on Circuits and Systems II: Analog. Digit. Signal Processing
46 (12), 1475–1486.Cox, D., Little, J., O’shea, D., 2007.
Ideals, varieties, and algorithms . Undergraduate Texts in Mathematics. Springer,New York, third edition.Decker, W., Greuel, G.-M., Pfister, G., Schoenemann, H., 2016.
SINGULAR 4.0.3. a computer al-gebra system for polynomial computations , FB Mathematik der Universitaet, D-67653 Kaiserslautern. .Decker, W., Lossen, C., 2006.
Computing in Algebraic Geometry . Algorithms and Computation in Mathematics.Springer Berlin Heidelberg.Fabia´nska, A., Quadrat, A., 2006. Applications of the Qullen-Suslin theorem to multidimensional systems theory.
In Gr¨obner Bases in Control Theory and Signal Processing , edited by Park, H. and Regensburger, G., Walter deGruyter, Berlin., 23–106.Fabia´nska, A., Quadrat, A., 2007.
A Maple implementation of a constructive version of the Quillen-Suslin Theorem . .Guan, J., Li, W., Ouyang, B., 2018. On rank factorizations and factor prime factorizations for multivariate polynomialmatrices. Journal of Systems Science and Complexity
31 (6), 1647–1658.Guan, J., Li, W., Ouyang, B., 2019. On minor prime factorizations for multivariate polynomial matrices.
Multidi-mensional Systems and Signal Processing
30, 493–502.Guiver, J., Bose, N., 1982. Polynomial matrix primitive factorization over arbitrary coefficient field and relatedresults.
IEEE Transactions on Circuits and Systems
29 (10), 649–657.Lin, Z., 1988. On matrix fraction descriptions of multivariable linear n-D systems.
IEEE Transactions on Circuitsand Systems
35 (10), 1317–1322.Lin, Z., 1993. On primitive factorizations for n-D polynomial matrices. In:
IEEE International Symposium on Circuitsand Systems . pp. 601–618.Lin, Z., 1999a. Notes on n-D polynomial matrix factorizations.
Multidimensional Systems and Signal Processing
10 (4), 379–393.Lin, Z., 1999b. On syzygy modules for polynomial matrices.
Linear Algebra and Its Applications
298 (1-3), 73–86. in, Z., 2001. Further results on n-D polynomial matrix factorizations. Multidimensional Systems and Signal Pro-cessing
12 (2), 199–208.Lin, Z., Bose, N., 2001. A generalization of Serre’s conjecture and some related issues.
Linear Algebra and ItsApplications
338 (1), 125–138.Lin, Z., Ying, J., Xu, L., 2001. Factorizations for n-D polynomial matrices.
Circuits, Systems, and Signal Processing
20 (6), 601–618.Liu, J., Li, D., Wang, M., 2011. On general factorizations for n-D polynomial matrices.
Circuits Systems and SignalProcessing
30 (3), 553–566.Liu, J., Li, D., Zheng, L., 2014. The Lin-Bose problem.
IEEE Transactions on Circuits and Systems II Express Briefs
61 (1), 41–43.Liu, J., Wang, M., 2010. Notes on factor prime factorizations for n-D polynomial matrices.
Multidimensional Systemsand Signal Processing
21 (1), 87–97.Liu, J., Wang, M., 2013. New results on multivariate polynomial matrix factorizations.
Linear Algebra and ItsApplications
438 (1), 87–95.Liu, J., Wang, M., 2015. Further remarks on multivariate polynomial matrix factorizations.
Linear Algebra and ItsApplications
465 (465), 204–213.Logar, A., Sturmfels, B., 1992. Algorithms for the Quillen-Suslin theorem.
Journal of Algebra
145 (1), 231–239.Lu, D., Ma, X., Wang, D., 2017. A new algorithm for general factorizations of multivariate polynomial matrices. In: proceedings of International Symposium on Symbolic and Algebraic Computation . pp. 277–284.Morf, M., Levy, B., Kung, S., 1977. New results in 2-D systems theory, part i: 2-D polynomial matrices, factorization,and coprimeness.
Proceedings of the IEEE
64 (6), 861–872.Park, H., 1995.
A computational theory of Laurent polynomial rings and multidimensional FIR systems . Ph.D. thesis,University of California at Berkeley.Pommaret, J., 2001. Solving Bose conjecture on linear multidimensional systems. In:
European Control Conference .IEEE, Porto, Portugal, pp. 1653–1655.Quillen, D., 1976. Projective modules over polynomial rings.
Inventiones mathematicae
36 (1), 167–171.Serre, J., 1955. Faisceaux alg´ebriques coh´erents.
Annals of Mathematics
61 (2), 197–278.Strang, G., 2010.
Linear algebra and its applications . Academic Press.Suslin, A., 1976. Projective modules over polynomial rings are free.
Soviet Math. Dokl.
17, 1160–1164.Wang, M., 2007. On factor prime factorization for n-D polynomial matrices.
IEEE Transactions on Circuits andSystems
54 (6), 1398–1405.Wang, M., Feng, D., 2004. On Lin-Bose problem.
Linear Algebra and Its Applications
390 (1), 279–285.Wang, M., Kwong, C., 2005. On multivariate polynomial matrix factorization problems.
Mathematics of Control,Signals, and Systems
17 (4), 297–311.Youla, D., Gnavi, G., 1979. Notes on n-dimensional system theory.
IEEE Transactions on Circuits and Systems
26 (2), 105–111.Youla, D., Pickel, P., 1984. The Quillen-Suslin theorem and the structure of n-dimensional elementary polynomialmatrices.
IEEE Transactions on Circuits and Systems
31 (6), 513–518.31 (6), 513–518.