On Factor Left Prime Factorization Problems for Multivariate Polynomial Matrices
aa r X i v : . [ c s . S C ] O c t On Factor Left Prime Factorization Problems for MultivariatePolynomial Matrices
Dong Lu a,b , Dingkang Wang c,d , Fanghui Xiao c,d, ∗ 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
Abstract
This paper is concerned with factor left prime factorization problems for multivariate polynomialmatrices without full row rank. We propose a necessary and su ffi cient condition for the existenceof factor left prime factorizations of a class of multivariate polynomial matrices, and then designan algorithm to compute all factor left prime factorizations if they exist. We implement thealgorithm on the computer algebra system Maple, and two examples are given to illustrate thee ff ectiveness of the algorithm. The results presented in this paper are also true for the existenceof factor right prime factorizations of multivariate polynomial matrices without full column rank. Keywords:
Multivariate polynomial matrices, Matrix factorization, Factor left prime (FLP),Column reduced minors, Free modules
1. Introduction
The factorization problems of multivariate polynomial matrices have attracted much attentionover the past decades because of their fundamental importance in multidimensional systems, cir-cuits, signal processing, controls, and other related areas (Bose, 1982; Bose et al., 2003). Up tonow, the factorization problems have been solved for univariate and bivariate polynomial matri-ces (Guiver and Bose, 1982; Morf et al., 1977). However, there are still many challenging openproblems for multivariate (more than two variables) polynomial matrix factorizations due to thelack of a mature polynomial matrix theory.Youla and Gnavi (1979) studied the basic structure of multidimensional systems theory, andproposed three types of factorizations for multivariate polynomial matrices: zero prime fac-torization, minor prime factorization and factor prime factorization. The existence problem ofzero prime factorizations for multivariate polynomial matrices with full rank first raised in (Lin,1999), and has been solved in (Pommaret, 2001; Wang and Feng, 2004). In recent years, the fac-torization problems of multivariate polynomial matrices without full rank deserve some attention.Lin and Bose (2001) studied a generalization of Serre’s conjecture, and they pointed out some ∗ Corresponding author
Email addresses: [email protected] (Dong Lu), [email protected] (Dingkang Wang), [email protected] (Fanghui Xiao) elationships between the existence of a zero prime factorization for a multivariate polynomialmatrix without full rank and its an arbitrary full rank submatrix.Wang and Kwong (2005) completely solved the existence problem of minor prime factoriza-tions for multivariate polynomial matrices with full rank, and proposed an e ff ective algorithm.Guan et al. (2019) extended the main result in (Wang and Kwong, 2005) to the case of non-fullrank. In order to study the existence problem of factor prime factorizations for multivariate poly-nomial matrices with full rank, Wang (2007) proposed the concept of regularity and obtained anecessary and su ffi cient condition. Guan et al. (2018) gave an algorithm to determine whether aclass of multivariate polynomial matrices without full rank has factor prime factorizations.Although some achievements have been made on the existence for factor prime factoriza-tions of some classes of multivariate polynomial matrices, factor prime factorizations are stillopen problems. Therefore, we focus on factor left prime factorization problems for multivariatepolynomial matrices without full row rank in this paper.The rest of the paper is organized as follows. In section 2, we introduce some basic conceptsand present the two major problems on factor left prime factorizations. We present in section3 a necessary and su ffi cient condition for the existence of factor left prime factorizations of aclass of multivariate polynomial matrices without full row rank. In section 4, we construct analgorithm and use two examples to illustrate the e ff ectiveness of the algorithm. We end withsome concluding remarks in section 5.
2. Preliminaries and Problems
We denote by k an algebraically closed field, z the n variables z , . . . , z n where n ≥
3. Let k [ z ]be the polynomial ring, and k [ z ] l × m be the set of l × m matrices with entries in k [ z ]. Throughoutthis paper, we assume that l ≤ m . In addition, we use “w.r.t.” to represent “with respect to”.For any given polynomial matrix F ∈ k [ z ] l × m , let rank( F ) and F T be the rank and the trans-posed matrix of F , respectively; if l = m , we use det( F ) to denote the determinant of F ; we denoteby ρ ( F ) the submodule of k [ z ] × m generated by the rows of F ; for each i with 1 ≤ i ≤ rank( F ), let d i ( F ) be the greatest common divisor of all the i × i minors of F ; let Syz( F ) be the syzygy moduleof F , i.e., Syz( F ) = { ~ v ∈ k [ z ] m × : F ~ v = ~ } . The following three concepts, which were first proposed in (Youla and Gnavi, 1979), play animportant role in multidimensional systems.
Definition 1.
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, then F issaid to be an minor left prime (MLP) matrix. If for any polynomial matrix factorization F = F F in which F ∈ k [ z ] l × l , F is necessarilya unimodular matrix, i.e., det( F ) is a nonzero constant, then F is said to be a factor leftprime (FLP) matrix. Let F ∈ k [ z ] m × l with m ≥ l , then a ZRP (MRP, FRP) matrix can be similarly defined. Notethat ZLP ⇒ MLP ⇒ FLP. Youla and Gnavi proved that when n =
1, the three concepts coincide;when n =
2, ZLP is not equivalent to MLP, but MLP is the same as FLP; when n ≥
3, theseconcepts are pairwise di ff erent.A factorization of a multivariate polynomial matrix is formulated as follows.2 efinition 2. Let F ∈ k [ z ] l × m with rank r and f is a divisor of d r ( F ) , where ≤ r ≤ l. F is saidto admit a factorization w.r.t. f if F can be factorized as F = G F (1) such that F ∈ k [ z ] r × m , G ∈ k [ z ] l × r with d r ( G ) = f . In particular, Equation (1) is said to be aZLP (MLP, FLP) factorization of F w.r.t. f if F is a ZLP (MLP, FLP) matrix. In order to state conveniently problems and main conclusions of this paper, we introduce thefollowing concepts and results.
Definition 3.
Let K be a submodule of k [ z ] × m , and J be an ideal of k [ z ] . We define K : J = { ~ u ∈ k [ z ] × m : J ~ u ⊆ K} , where J ~ u is the set { f ~ u : f ∈ J } . Obviously,
K ⊆ K : J . Let I ⊂ k [ z ] be another ideal, it is easy to show that K : ( I J ) = ( K : I ) : J . (2)Equation (2) is a simple generalization of Proposition 10 in subsection 4, Zariski closure andquotients of ideals in (Cox et al., 2007). For convention, we write K : h f i as K : f for any f ∈ k [ z ]. Definition 4.
Let K be a k [ z ] -module. The torsion submodule of K is defined as Torsion( K ) = { ~ u ∈ K : ∃ f ∈ k [ z ] \{ } such that f ~ u = ~ } . We refer to (Eisenbud, 2013) for more details about the above two concepts. Let K , K betwo k [ z ]-modules, we define K / K = { ~ u + K : ~ u ∈ K } . Liu and Wang (2015) established arelationship between Definition 3 and Definition 4. Lemma 5.
Let F ∈ k [ z ] l × m be of full row rank, d = d l ( F ) and K = ρ ( F ) . Then ( K : d ) / K = Torsion( k [ z ] × m / K ) . Moreover, Liu and Wang further extended the Youla’s MLP lemma, which had been used togive another proof of the Serre’s problem.
Lemma 6.
Let F ∈ k [ z ] l × m be of full row rank and d = d l ( F ) . Then for each i = , . . . , n, thereexists V i ∈ k [ z ] m × l such that FV i = d ϕ i I l × l , where ϕ i is nonzero and independent of z i . Guan et al. (2018) proved the following lemma, which is similar to the above result.
Lemma 7.
Let G ∈ k [ z ] l × r be of full column rank with l ≥ r, and g be an arbitrary r × r minorof G . Then there exists G ′ ∈ k [ z ] r × l such that G ′ G = g I r × r . In order to study the properties of multivariate polynomial matrices, Lin (1988) and Sule(1994) introduced the following important concept.
Definition 8.
Let F ∈ k [ z ] l × m with rank r, where ≤ r ≤ l. For any given 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 yields a j = d i ( F ) · b j , j = , . . . , β. Then, b , . . . , b β are called all the i × i reduced minors of F . Lemma 9.
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 full columnrank submatrices of F , where 1 ≤ η ≤ (cid:16) mr (cid:17) . According to Lemma 9, it follows that ¯ F , . . . , ¯ F η have the same r × r reduced minors. Based on this phenomenon, we give the following conceptwhich was first proposed in (Lin and Bose, 2001). Definition 10.
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 all the r × r column reduced minors of F . The above concept will play an important role in this paper. Obviously, the calculationamount of all the r × r column reduced minors of F is much less than that of all the r × r reduced minors of F in general. Lemma 11.
Let U ∈ k [ z ] l × m be a ZLP matrix, where l < m. Then there exists a ZRP matrix V ∈ k [ z ] m × l such that UV = I l × l . Moreover, Syz( U ) is a free submodule of k [ z ] m × with rankm − l. The above result is called the Quillen-Suslin theorem. In order to solve the problem whetherany finitely generated projective module over a polynomial ring is free, Quillen (1976) and Suslin(1976) solved the problem positively and independently.Using the Quillen-Suslin theorem, Pommaret (2001) and Wang and Feng (2004) solved theLin-Bose conjecture.
Lemma 12.
Let F ∈ k [ z ] l × m be of full row rank, where l < m. If all the l × l reduced minors of F generate k [ z ] , then F has a ZLP factorization. Let F ∈ k [ z ] l × m be of full row rank, and f be a divisor of d l ( F ). In order to study a factorizationof F w.r.t. f , Wang (2007) introduced the concept of regularity. f is said to be regular w.r.t. F if and only if d l ([ f I l × l F ]) = f up to multiplication by a nonzero constant. Then, Wang obtainedthe following result. Lemma 13.
Let F ∈ k [ z ] l × m be of full row rank, and f be regular w.r.t. F . Then F has afactorization w.r.t. f if and only if ρ ( F ) : f is a free module of rank l.2.2. Problems According to Lemma 13, Wang proposed a necessary and su ffi cient condition to verifywhether F has a FLP factorization w.r.t. f . After that, Guan et al. (2018) considered the caseof multivariate polynomial matrices without full row rank. When f satisfies a special property,they obtained a necessary condition that F has a factorization w.r.t. f , and designed an algorithmto compute all FLP factorizations of F if they exist. In this paper we will further consider thefollowing two problems concerning FLP factorizations.4 roblem 14. Let F ∈ k [ z ] l × m with rank r, and f be a divisor of d r ( F ) , where ≤ r < l. Determinewhether F has a FLP factorization w.r.t. f . Problem 15.
Let F ∈ k [ z ] l × m with rank r, where ≤ r < l. Constructing an algorithm to computeall FLP factorizations of F . Youla and Gnavi used an example to show that it is very di ffi cult to judge whether a mul-tivariate polynomial matrix is a FLP matrix. Hence, Problem 14 and Problem 15 may be verydi ffi cult in general. In this paper, we will give partial solutions to the above two problems.
3. Main Results
Let F ∈ k [ z ] l × m with rank r , and f be a divisor of d r ( F ), where 1 ≤ r < l . We use thefollowing lemma to illustrate that all the r × r column reduced minors of F play an important rolein a factorization of F w.r.t. f . Lemma 16.
Let F ∈ k [ z ] l × m with rank r, f be a divisor of d r ( F ) , and c , . . . , c ξ be all the r × rcolumn reduced minors of F , where ≤ r < l. If there exist G ∈ k [ z ] l × r and F ∈ k [ z ] r × m suchthat F = G F with d r ( G ) = f , then I r ( G ) = h f c , . . . , f c ξ i .Proof. Since F is a matrix with rank r , there exists a full row rank matrix A ∈ k [ z ] ( l − r ) × l such that AF = ( l − r ) × m . Let ¯ F ∈ k [ z ] l × r be an arbitrary full column rank submatrix of F , then A ¯ F = ( l − r ) × r .Based on Lemma 9, all the r × r reduced minors of A are c , . . . , c ξ . It follows from rank( F ) ≤ min { rank( G ) , rank( F ) } that G is a full column rank matrix and F is a full row rank matrix.Then AG F = ( l − r ) × m implies that AG = ( l − r ) × r . Using Lemma 9 again, all the r × r reducedminors of G are c , . . . , c ξ . Consequently, I r ( G ) = h f c , . . . , f c ξ i since d r ( G ) = f .Now, we give the first main result in this paper. Theorem 17.
Let F ∈ k [ z ] l × m with rank r, f be a divisor of d r ( F ) and c , . . . , c ξ be all the r × rcolumn reduced minors of F , where ≤ r < l. Let d = d r ( F ) and K = ρ ( F ) , then the followingare equivalent: F has a factorization w.r.t. f ; there exists F ∈ k [ z ] r × m with full row rank such that d r ( F ) = df and K ⊆ ρ ( F ) ⊆ K : h f c , . . . , f c ξ i .Proof. →
2. Suppose that F has a factorization w.r.t. f . Then there exist G ∈ k [ z ] l × r and F ∈ k [ z ] r × m such that F = G F with d r ( G ) = f . Clearly, K ⊆ ρ ( F ). From d r ( F ) = d r ( G ) d r ( F )we have d r ( F ) = df . According to Lemma 16, I r ( G ) = h f c , . . . , f c ξ i . Let g be any r × r minorof G , then there exists G ′ ∈ k [ z ] r × l such that G ′ G = g I r × r by Lemma 7. Multiplying both leftsides of F = G F by G ′ , we get G ′ F = G ′ G F = g F . This implies that g · ρ ( F ) ⊆ K . Notingthat g is an arbitrary r × r minor of G , we obtain ρ ( F ) ⊆ K : I r ( G ) = K : h f c , . . . , f c ξ i .2 →
1. Thanks to
K ⊆ ρ ( F ), there exists G ∈ k [ z ] l × r such that F = G F . It follows from d r ( F ) = d r ( G ) d r ( F ) that d r ( G ) = f . Then, F has a factorization w.r.t. f .Although Theorem 17 gives a necessary and su ffi cient condition for F to have a factorizationw.r.t. f , it is di ffi cult to find a full row rank matrix F ∈ k [ z ] r × m that satisfies d r ( F ) = df and K ⊆ ρ ( F ) ⊆ K : h f c , . . . , f c ξ i . Next, we will further study the relationship between ρ ( F ) and ρ ( F ). 5 heorem 18. Let F ∈ k [ z ] l × m with rank r, f be a divisor of d r ( F ) and c , . . . , c ξ be all the r × rcolumn reduced minors of F , where ≤ r < l. Suppose there exist G ∈ k [ z ] l × r and F ∈ k [ z ] r × m such that F = G F with d r ( G ) = f . Let d = d r ( F ) , K = ρ ( F ) and K = ρ ( F ) , then thefollowing are equivalent:
1. ( K : df ) / K ;
2. ( K : h dc , . . . , dc ξ i ) / K ;
3. Torsion( k [ z ] × m / K ) .Proof. It follows from rank( F ) ≤ min { rank( G ) , rank( F ) } that F is a full row rank matrix. Since d r ( F ) = d r ( G ) d r ( F ), we have d r ( F ) = df . It is apparent from Lemma 5 that( K : df ) / K = Torsion( k [ z ] × m / K ) . (3)If the following equation K : df = K : h dc , . . . , dc ξ i (4)holds, then ( K : df ) / K and ( K : h dc , . . . , dc ξ i ) / K are obviously equivalent.We first verify K : df ⊆ K : h dc , . . . , dc ξ i . Proceeding as in the proof of 1 → K ⊆ K : h f c , . . . , f c ξ i . (5)Using Equation (2), we can derive K : df ⊆ ( K : h f c , . . . , f c ξ i ) : df = K : h dc , . . . , dc ξ i . (6)Next we show K : h dc , . . . , dc ξ i ⊆ K : df . For any vector ~ u ∈ K : h dc , . . . , dc ξ i = T ξ j = ( K : dc j ), there exists ~ v j ∈ k [ z ] × l such that dc j ~ u = ~ v j F = ~ v j G F , j = , . . . , ξ. (7)Using Lemma 6, for each i = , . . . , n , there exists V i ∈ k [ z ] m × r such that F V i = df ϕ i I r × r , (8)where ϕ i is nonzero and independent of z i . Combining Equation (7) and Equation (8), we seethat dc j ~ u V i = ~ v j G F V i = ~ v j G ( df ϕ i I r × r ) = df ϕ i ~ v j G . (9)As gcd( ϕ , . . . , ϕ n ) =
1, we have dc j | df ~ v j G . This implies that ~ v j G fc j is a polynomial vector.Then, it follows from Equation (7) that df ~ u = ~ v j G f c j F , j = , . . . , ξ. (10)Thus, ~ u ∈ K : df , and we infer that K : h dc , . . . , dc ξ i ⊆ K : df .Consequently, ( K : df ) / K = ( K : h dc , . . . , dc ξ i ) / K .6n Theorem 18, we obtain K : df = K : h dc , . . . , dc ξ i . Naturally, we consider under whatconditions K and K : h f c , . . . , f c ξ i are equal. Now, we propose the following conclusion. Theorem 19.
Let F ∈ k [ z ] l × m with rank r, f be a divisor of d r ( F ) and c , . . . , c ξ be all the r × rcolumn reduced minors of F , where ≤ r < l. Suppose there exist G ∈ k [ z ] l × r and F ∈ k [ z ] r × m such that F = G F with d r ( G ) = f . Let d = d r ( F ) , K = ρ ( F ) and K = ρ ( F ) . If gcd( f , df ) = ,then K = K : h f c , . . . , f c ξ i and K : h f c , . . . , f c ξ i is a free module of rank r. The above theorem is a generalization of Theorem 3.11 in (Guan et al., 2018). The proofof Theorem 19 is basically the same as that of Theorem 3.11, except that we explicitly give asystem of generators of I r ( G ). Hence, the proof is omitted here. Evidently, the calculationamount of ρ ( F ) = ρ ( F ) : h f b , . . . , f b β i in Theorem 3.11 is much larger than that of ρ ( F ) = ρ ( F ) : h f c , . . . , f c ξ i in Theorem 19.Suppose gcd( f , df ) =
1. Let K : h f c , . . . , f c ξ i be a free module of rank r , and a free basis ofthe module constitutes F ∈ k [ z ] r × m . Then, ρ ( F ) = K : h f c , . . . , f c ξ i . Given K ⊆ ρ ( F ), thereexists G ∈ k [ z ] l × r such that F = G F with d r ( G ) = f ′ , where f ′ is a divisor of d . Notice that f and f ′ may be di ff erent. The condition that K : h f c , . . . , f c ξ i is a free module of rank r isonly a necessary condition for the existence of a factorization of F w.r.t. f . In order to study therelationship between f ′ and f , we first introduce a result in (Liu and Wang, 2015). Lemma 20.
Let F ∈ k [ z ] l × m be of full row rank, d = d l ( F ) and K = ρ ( F ) . If there exists a divisorf of d such that K : f = K , then f is a constant. Now, we can draw the following conclusion.
Proposition 21.
Let F ∈ k [ z ] l × m with rank r, and c , . . . , c ξ be all the r × r column reducedminors of F , where ≤ r < l. Let K = ρ ( F ) , d = d r ( F ) be a square-free polynomial and f bea divisor of d. Suppose K = K : h f c , . . . , f c ξ i is a free module of rank r and F ∈ k [ z ] r × m iscomposed of a free basis of K . Then, there is no a proper divisor f ′ of f such that F = G F ,where G ∈ k [ z ] l × r with d r ( G ) = f ′ .Proof. Note that
K ⊆ K , there exists G ∈ k [ z ] l × r such that F = G F with d r ( G ) = f ′ , where f ′ is a divisor of d . Since d is a square-free polynomial, gcd( f ′ , df ′ ) =
1. According to Theorem19, it follows that K = K : h f ′ c , . . . , f ′ c ξ i , i.e., K : h f c , . . . , f c ξ i = K : h f ′ c , . . . , f ′ c ξ i . (11)Assume that f ′ is a proper divisor of f . It can easily be seen from Equation (11) that K : ff ′ = K . (12)Because d r ( F ) = df ′ , we have ff ′ | d r ( F ). Based on Lemma 20, ff ′ is a constant. This contradictsthe fact that f ′ is a proper divisor of f .Before giving a new necessary and su ffi cient condition for the existence of a factorization of F w.r.t. f , we present the following result. Lemma 22.
Let F ∈ k [ z ] l × m with rank r, and c , . . . , c ξ be all the r × r column reduced minorsof F , where ≤ r < l. Then the following are equivalent: there exist U ∈ k [ z ] l × r and F ∈ k [ z ] r × m such that F = UF with U being a ZRP matrix; h c , . . . , c ξ i = k [ z ] .Proof. →
2. Suppose there exist U ∈ k [ z ] l × r and F ∈ k [ z ] r × m such that F = UF , where U is a ZRP matrix. Using Lemma 16, c , . . . , c ξ are all the r × r reduced minors of U . Then, h c , . . . , c ξ i = k [ z ] since U is a ZRP matrix.2 →
1. Because rank( F ) = r , there exists a full row rank matrix H ∈ k [ z ] ( l − r ) × l such that HF = ( l − r ) × m . (13)According to Lemma 9, c , . . . , c ξ are all the ( l − r ) × ( l − r ) reduced minors of H . Assume that h c , . . . , c ξ i = k [ z ]. By Lemma 12, H has a ZLP factorization H = GH , (14)where G ∈ k [ z ] ( l − r ) × ( l − r ) , and H ∈ k [ z ] ( l − r ) × l is a ZLP matrix. Let ~ v ∈ Syz( H ), then H ~ v = GH ~ v = ~
0. Since G is a full column rank matrix, H ~ v = ~
0. This implies that ~ v ∈ Syz( H ). Let ~ u ∈ Syz( H ), it is obvious that ~ u ∈ Syz( H ). It follows thatSyz( H ) = Syz( H ) . (15)Thus we conclude that Syz( H ) is a free module of rank r by the Quillen-Suslin theorem.Suppose that U ∈ k [ z ] l × r is composed of a free basis of Syz( H ). It follows from HU = ( l − r ) × r that all the r × r reduced minors of U generate k [ z ]. Using Lemma 12 again, there exist U ∈ k [ z ] l × r and G ∈ k [ z ] r × r such that U = U G (16)with U being a ZRP matrix. Since G is a full row rank matrix, from HU G = ( l − r ) × r we have HU = ( l − r ) × r . (17)This implies that ρ ( U T1 ) ⊆ ρ ( U T ) . (18)Using d r ( U ) = d r ( U )det( G ), we get d r ( U ) = δ det( G ), where δ is a nonzero constant. Ifdet( G ) ∈ k [ z ] \ k , then Equation (16) implies that ρ ( U T ) ( ρ ( U T1 ) . (19)This leads to a contradiction. Thus, det( G ) is a nonzero constant. Consequently, we infer that U is a ZRP matrix.Equation (13) implies that the columns of F belong to Syz( H ), then there exists F ∈ k [ z ] r × m such that F = UF . (20)Now, we give the second main result in this paper. Theorem 23.
Let F ∈ k [ z ] l × m with rank r, and c , . . . , c ξ be all the r × r column reduced minorsof F , where ≤ r < l. Let K = ρ ( F ) , d = d r ( F ) and f be a divisor of d with gcd( f , df ) = . If h c , . . . , c ξ i = k [ z ] , then the following are equivalent: F has a factorization w.r.t. f ; K : f is a free module of rank r.Proof. →
2. Suppose that F has a factorization w.r.t. f . Then there exist G ∈ k [ z ] l × r and F ∈ k [ z ] r × m such that F = G F with d r ( G ) = f . According to Theorem 19, ρ ( F ) = K : h f c , . . . , f c ξ i . It follows from h c , . . . , c ξ i = k [ z ] that h f c , . . . , f c ξ i = h f i . Then, ρ ( F ) = K : f . As F is a full row rank matrix, K : f is a free module of rank r .2 →
1. Since h c , . . . , c ξ i = k [ z ], by Lemma 22 we obtain F = UF ′ , (21)where U ∈ k [ z ] l × r is a ZRP matrix and F ′ ∈ k [ z ] r × m . Without loss of generality, we assumethat d r ( U ) =
1. Clearly, ρ ( F ) ⊆ ρ ( F ′ ). Based on the Quillen-Suslin theorem, there is a ZLPmatrix V ∈ k [ z ] r × l such that VU = I r × r . Then, F ′ = VF . This implies that ρ ( F ′ ) ⊆ ρ ( F ). Thus, ρ ( F ′ ) = K , d r ( F ′ ) = d r ( F ) and ρ ( F ′ ) : f is a free module of rank r . Since gcd( f , df ) = f isregular w.r.t. F ′ . By Lemma 13, there exist G ′ ∈ k [ z ] r × r and F ∈ k [ z ] r × m such that F ′ = G ′ F (22)with det( G ′ ) = f . By substituting Equation (22) into Equation (21), we get F = ( UG ′ ) F . (23)Let G = UG ′ , then d r ( G ) = d r ( U )det( G ′ ) = f . Thus F has a factorization w.r.t. f . Remark 24.
Wang (2007) proved that f is regular w.r.t. F ′ if gcd( f , df ) = . Let F ∈ k [ z ] l × m with rank r and f be a divisor of d r ( F ), where 1 ≤ r < l . We define thefollowing set: M ( f ) = { h ∈ k [ z ] : f | h and h | d r ( F ) } . Now, we give a partial solution to Problem 14.
Theorem 25.
Let F ∈ k [ z ] l × m with rank r, and c , . . . , c ξ be all the r × r column reduced minorsof F , where ≤ r < l. Let K = ρ ( F ) , d = d r ( F ) and f be a divisor of d. Suppose every h ∈ M ( f ) satisfies gcd( h , dh ) = and h c , . . . , c ξ i = k [ z ] , then the following are equivalent: F has a FLP factorization w.r.t. f ; K : f is a free module of rank r, but K : h is not a free module of rank r for everyh ∈ M ( f ) \ { f } . Remark 26.
With the help of Theorem 23, the proof of Theorem 25 is similar to that of Theorem3.2 in (Wang, 2007), and is omitted here.
In the above theorem, we need to verify whether a submodule of k [ z ] × m is a free module ofrank r . The traditional method is to calculate the r -th Fitting ideal of the submodule. We refer to(Cox et al., 2005; Eisenbud, 2013; Greuel and Pfister, 2002) for more details. Next, we will givea simpler verification method. 9 roposition 27. Let F ∈ k [ z ] l × m with rank r, and J ⊂ k [ z ] be a nonzero ideal, where ≤ r < l.Suppose F ∈ k [ z ] s × m is composed of a system of generators of ρ ( F ) : J, then the following areequivalent: ρ ( F ) : J is a free module of rank r; all the r × r column reduced minors of F generate k [ z ] .Proof. It is evident that ρ ( F ) : J = ρ ( F ). According to Proposition 3.14 in (Guan et al., 2018),the rank of ρ ( F ) : J is r . This implies that rank( F ) = r and s ≥ r .1 →
2. Suppose that ρ ( F ) : J is a free module of rank r . Let F ∈ k [ z ] r × m be composed ofa free basis of ρ ( F ) : J , then ρ ( F ) = ρ ( F ). On the one hand, ρ ( F ) ⊆ ρ ( F ) implies that thereexists G ∈ k [ z ] s × r such that F = G F . On the other hand, it follows from ρ ( F ) ⊆ ρ ( F ) thatthere exists G ∈ k [ z ] r × s such that F = G F . Combining the above two equations, we have F = ( G G ) F . Because F is a full row rank matrix, we obtain I r × r = G G . According to theBinet-Cauchy formula, all the r × r minors of G generate k [ z ]. Therefore, G is a ZRP matrix.Based on Lemma 22, all the r × r column reduced minors of F generate k [ z ].2 →
1. There are two cases. First, s > r . Using Lemma 22, there exist F ∈ k [ z ] r × m anda ZRP matrix U ∈ k [ z ] s × r such that F = UF . It follows from the proof of 2 → ρ ( F ) = ρ ( F ). Since F is a full row rank matrix, ρ ( F ) : J is a free module of rank r .Second, s = r . In this situation, F is a full row rank matrix. This implies that ρ ( F ) : J is a freemodule of rank r . Obviously, all the r × r column reduced minors of F are only one polynomialwhich is the constant 1, and generate k [ z ]. In summary, ρ ( F ) : J is a free module of rank r .
4. Algorithm and Examples
Before solving Problem 15, we make the following analysis on the main results obtained insection 3. We first construct a polynomial matrix set of k [ z ] l × m as follows: M = { F ∈ k [ z ] l × m : d r ( F ) is a square-free polynomial } , where r = rank( F ). Let F ∈ M , d = d r ( F ), K = ρ ( F ), f be an arbitrary divisor of d , and c , . . . , c ξ be all the r × r column reduced minors of F , where 1 ≤ r < l . There are two cases as follows.First, h c , . . . , c ξ i = k [ z ]. According to Theorem 23, F has a factorization w.r.t. f if and onlyif K : f is a free module of rank r . Since f is an arbitrary divisor of d , we can compute all matrixfactorizations of F . After that, we obtain all FLP factorizations of F by Theorem 25.Second, h c , . . . , c ξ i , k [ z ]. We only get a necessary condition for the existence of a factor-ization of F w.r.t. f in Theorem 19. Nevertheless, we can get all factorizations of F . The specificprocess is as follows. Let f , . . . , f s be all di ff erent divisors of d and K j = K : h f j c , . . . , f j c ξ i ,then we verify whether K j is a free module of rank r , where j = , . . . , s . For each j , one of thefollowing three cases holds:1. K j is not a free module of rank r , then F has no factorization w.r.t. f j ;2. K j is a free module of rank r , and a free basis of K j constitutes F j ∈ k [ z ] r × m ,2.1 if d r ( F j ) = df j , then F has a factorization w.r.t. f j ;2.2 if d r ( F j ) , df j , then F has a factorization w.r.t. f i , where f i ∤ f j .10et F = G i F i = · · · = G i t F i t be all di ff erent factorizations of F and K i j = ρ ( F i j ), where G i j ∈ k [ z ] l × r , F i j ∈ k [ z ] r × m , j = , . . . , t and 0 ≤ t ≤ s ( t = F has no factorizations).For each K i j , if there does not exist j ′ such that K i j ( K i j ′ , then F = G i j F i j is a FLP factorizationof F . The reason is as follows. Assume that there exist G ∈ k [ z ] r × r and F ∈ k [ z ] r × m such that F i j = G F . If det( G ) ∈ k [ z ] \ k , then K i j ( ρ ( F ). It can be seen that F = ( G i j G ) F is afactorization of F and it is di ff erent from F = G i j F i j . This contradicts the fact that there exists no j ′ such that K i j ( K i j ′ . Then, det( G ) is a nonzero constant and F i j is a FLP matrix.According to the above analysis, we now give a partial solution to Problem 15. We constructthe following algorithm to compute all FLP factorizations for F ∈ M .Before proceeding further, let us remark on Algorithm 1.(1) In step 14 and step 26, we need to compute free bases of free submodules in k [ z ] × m .Fabia´nska and Quadrat (2007) first designed a Maple package, which is called QUILLEN-SUSLIN, to implement the Quillen-Suslin theorem. At the same time, they implementedan algorithm for computing free bases of free submodules in this package. Based on thisfact, Algorithm 1 is implemented on Maple. For interested readers, more examples can begenerated by the codes at: .(2) In step 8 and step 20, we need to compute a system of generators of K : J , where K ⊂ k [ z ] × m and J is a nonzero ideal. Wang and Kwong (2005) proposed an algorithm tocompute K : J , and we have implemented this algorithm on Maple.(3) In step 9 and step 21, if F ′ i is a full row rank matrix, then ρ ( F ′ i ) is a free module of rank r and we do not need to compute a reduced Gr¨obner basis of all the r × r column reducedminors of F ′ i ; otherwise, we need to use Proposition 27 to determine whether K : J is afree module of rank r .(4) In step 20, ρ ( F ) : ( f i G ) = ρ ( F ) : h f i c , . . . , f i c ξ i since G is a reduced Gr¨obner basis of h c , . . . , c ξ i . This can help us reduce some calculations.(5) In step 15 and step 27, we need to compute G i ∈ k [ z ] l × r such that F = G i F i . Lu et al. (2020)designed a Maple package, which is called poly-matrix-equation, for solving multivariatepolynomial matrix Diophantine equations. We use this package to compute G i .(6) In step 15, Theorem 23 can guarantee that d r ( G i ) = f i . In step 27, we can not ensure that d r ( G i ) = f i . Proposition 21 only tell us that there is no a proper divisor f ′ i of f i such that d r ( G i ) = f ′ i . Hence, we need to compute d r ( G i ).(7) In step 25 and step 29, we can use Gr¨obner bases to verify the inclusion relationship oftwo submodules of k [ z ] × m .(8) In step 17, the element ( F ′ i , f i ) is also deleted since f i divides itself. Similarly, the element( F ′ i , f i ) in step 29 is also deleted since ρ ( F ′ i ) ⊆ ρ ( F ′ i ).(9) In fact, we can obtain all factorizations of F by making appropriate modifications to Algo-rithm 1. 11 lgorithm 1: FLP factorization algorithm
Input : F ∈ M , the rank r of F and d r ( F ). Output: all FLP factorizations of F . begin P : = ∅ and W : = ∅ ; compute all di ff erent divisors f , . . . , f s of d r ( F ); compute all the r × r column reduced minors c , . . . , c ξ of F ; compute a reduced Gr¨obner basis G of h c , . . . , c ξ i ; if G = { } then for i from to s do compute a system of generators of ρ ( F ) : f i , and use all the elements in thesystem to constitute a matrix F ′ i ∈ k [ z ] s i × m ; if the reduced Gr¨obner basis of all the r × r column reduced minors of F ′ i is { } then P : = P ∪ { ( F ′ i , f i ) } ; while P , ∅ do select any element ( F ′ i , f i ) from P ; if there is no other elements ( F ′ j , f j ) ∈ P such that f i | f j then compute a free basis of ρ ( F ′ i ), and use all the elements in the basis toconstitute a matrix F i ∈ k [ z ] r × m ; compute a matrix G i ∈ k [ z ] l × r such that F = G i F i ; W : = W ∪ { ( G i , F i , f i ) } ; delete all elements ( F ′ t , f t ) that satisfy f t | f i from P ; else for i from to s do compute a system of generators of ρ ( F ) : ( f i G ), and use all the elements inthe system to constitute a matrix F ′ i ∈ k [ z ] s i × m ; if the reduced Gr¨obner basis of all the r × r column reduced minors of F ′ i is { } then P : = P ∪ { ( F ′ i , f i ) } ; while P , ∅ do select any element ( F ′ i , f i ) from P ; if there is no other elements ( F ′ j , f j ) ∈ P such that ρ ( F ′ i ) ( ρ ( F ′ j ) then compute a free basis of ρ ( F ′ i ), and use all the elements in the basis toconstitute F i ∈ k [ z ] r × m ; compute a matrix G i ∈ k [ z ] l × r such that F = G i F i with d r ( G i ) = f ′ i ; W : = W ∪ { ( G i , F i , f ′ i ) } ; delete all elements ( F ′ t , f t ) that satisfy ρ ( F ′ t ) ⊆ ρ ( F ′ i ) from P ; return W . 12 .2. Examples We first use the example in (Guan et al., 2018) to illustrate the calculation process of Algo-rithm 1.
Example 28.
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 be a multivariate polynomial matrix in C [ z , z , z ] × , where z > z > z and C is the complexfield.It is easy to compute that the rank of F is 2, and d ( F ) = ( z − z . Since d ( F ) is a square-free polynomial, F ∈ M . Then, we can use Algorithm 1 to compute all FLP factorizations of F .The input of Algorithm 1 are F , r = d ( F ) = ( z − z .Let P = ∅ and W = ∅ . All di ff erent divisors of d ( F ) are: f = f = z − f = z and f = ( z − z . All the 2 × F are: c = c = z and c = − z .The reduced Gr¨obner basis of h c , c , c i w.r.t. the degree reverse lexicographic order is G = { } .Now, we use the steps from 7 to 17 to compute all FLP factorizations of F .(1) When i =
1, we first compute a system of generators of ρ ( F ) : f and the system is { [ z z − z , , z + , [0 , z z − z , z − z + } . Let F ′ = " z z − z z + z z − z z − z + . Since ρ ( F ′ ) = ρ ( F ) : f and F ′ is a full row rank matrix, ρ ( F ) : f is a free module of rank 2.(2) When i =
2, a system of generators of ρ ( F ) : f is { [0 , z , z − , [ z z − z , , z + } .Let F ′ = " z z − z z − z z + . Since ρ ( F ′ ) = ρ ( F ) : f and F ′ is a full row rank matrix, ρ ( F ) : f is a free module of rank 2.(3) When i =
3, a system of generators of ρ ( F ) : f is { [ z z − z , , z + , [0 , z z − z , z − z + , [ z − z + z − , − z z − z + z + , } . Let F ′ = z z − z z + z z − z z − z + z − z + z − − z z − z + z + . All the 2 × F ′ are ( z − , − z , z +
1. Since h ( z − , − z , z + i , C [ z , z , z ], ρ ( F ) : f is not a free module of rank 2.(4) When i =
4, a system of generators of ρ ( F ) : f is { [0 , z , z − , [ z z − z , , z + , [ z − z + , − z − , } . Let F ′ = z z − z z − z z + z − z + − z − . All the 2 × F ′ are z − , z , z +
1. Since h z − , z , z + i , C [ z , z , z ], ρ ( F ) : f is not a free module of rank 2.13hen, P = { ( F ′ , f ) , ( F ′ , f ) } . Since f is a proper multiple of f , F has a FLP factorizationw.r.t. f . Obviously, the rows of F ′ constitute a free basis of ρ ( F ) : f . Let F = F ′ , we computea polynomial matrix G ∈ C [ z , z , z ] × such that F = G F = z − z z − z z " z z − z z − z z + , where d ( G ) = f and F is a FLP matrix. Then, W = { ( G , F , f ) } . Remark 29.
Since h c , c , c i = h i , we can use Theorem 25 to compute all FLP factorizationsof F . The above calculation process is simpler than that of Example 3.20 in (Guan et al., 2018).Obviously, Algorithm 1 is more e ffi cient than the algorithm proposed in (Guan et al., 2018). Example 30.
Let F = z z z z z z + z z z z z z z z z be a multivariate polynomial matrix in C [ z , z , z ] × , where z > z > z and C is the complexfield.It is easy to compute that the rank of F is 2, and d ( F ) = z z z . Since d ( F ) is a square-freepolynomial, F ∈ M . Then, we can use Algorithm 1 to compute all FLP factorizations of F . Theinput of Algorithm 1 are F , r = d ( F ) = z z z .Let P = ∅ and W = ∅ . All di ff erent divisors of d ( F ) are: f = f = z , f = z , f = z , f = z z , f = z z , f = z z and f = z z z . All the 2 × F are: c = z , c = z and c = z z . The reduced Gr¨obner basis of h c , c , c i w.r.t. the degree reverselexicographic order is G = { z , z } . Now, we use the steps from 19 to 29 to compute all FLPfactorizations of F .Let K i = ρ ( F ) : h f i c , f i c , f i c i , where i = , . . . ,
8. Since G is a Gr¨obner basis of h c , c , c i ,for each i we have K i = ρ ( F ) : h f i c , f i c i = ( ρ ( F ) : f i c ) ∩ ( ρ ( F ) : f i c ).(1) When i =
1, the systems of generators of ρ ( F ) : z and ρ ( F ) : z are { [ z z , , z z ] , [0 , z z , z ] , [ − z z , z z , } and { [ z z , , z z ] , [0 , z z , z ], [0 , z , z z ] } , respectively. Then, a sys-tem of generators of K is { [ z z , , z z ] , [0 , z z , z ] , [ − z z z , z z z , } . Let F ′ = z z z z z z z − z z z z z z . It is easy to compute that all the 2 × F ′ are 1 , z z , z . Since h , z z , z i = C [ z , z , z ], K is a free module of rank 2.(2) When i =
2, the systems of generators of ρ ( F ) : z and ρ ( F ) : z z are { [ z z , , z z ] , [0 , z z , z ] , [ z z , − z z , } and { [ z z , , z z ] , [0 , z , z ], [ z z , − z z , } , respectively.Then, a system of generators of K is { [ z z , , z z ] , [0 , z z , z ] , [ z z , − z z , } . F ′ = z z z z z z z z z − z z . It is easy to compute that all the 2 × F ′ are z , − z , − z . Since h z , − z , − z i , C [ z , z , z ], K is not a free module of rank 2.(3) According to the above same steps, we have that the systems of generators of K , . . . , K are { [ z , , z ] , [0 , z z , z ] } , { [0 , z , z ] , [ z z , , z z ] } , { [ − z , z , , [ z , , z ] } , { [0 , z , z ] , [ z , − z , } , { [ z , , z ] , [0 , z , z ] } and { [0 , z , z ] , [ − , , } , respectively. Let F ′ i ∈ C [ z , z , z ] × be composed of the above system of generators of K i , where i = , . . . ,
8. Foreach i , it is easy to compute that rank( F ′ i ) =
2. This implies that F ′ i is a full row rank matrix.Then, K i = ρ ( F ′ i ) is a free module of rank 2. Then, we have P = { ( F ′ , f ) , ( F ′ , f ) , . . . , ( F ′ , f ) } . (4) Since ρ ( F ′ i ) ( ρ ( F ′ ) for each 1 ≤ i ≤ i , F has only one FLP factorization.Since F ′ = " z z − is a full row rank matrix, the rows of F ′ constitute a free basis of K = ρ ( F ′ ). Let F = F ′ , wecompute a polynomial matrix G ∈ C [ z , z , z ] × such that F = G F = z + z − z z z − z z z z " z z − , where F is a FLP matrix. It is easy to compute that d ( G ) = f . Then, W = { ( G , F , f ) } .
5. Concluding Remarks
In this paper we have studied two FLP factorization problems for multivariate polynomialmatrices without full row rank. As we all know, FLP factorizations are still open problems sofar. In order to solve some special situations, we have introduced the concept of column reducedminors. Then, we have proved a theorem which provides a necessary and su ffi cient conditionfor a class of multivariate polynomial matrices without full row rank to have FLP factorizations.Moreover, we have given a simple method to verify whether a submodule of k [ z ] × m is a freemodule by using column reduced minors of polynomial matrices. Compared with the traditionalmethod, the new method is more e ffi cient. Based on our results, we have also proposed analgorithm for FLP factorizations and have implemented it on the computer algebra system Maple.Two examples have been given to illustrate the e ff ectiveness of the algorithm.Let F ∈ k [ z ] l × m , every full column rank submatrix of F is a square matrix if rank( F ) = l . Inthis case, all the l × l column reduced minors of F are only one polynomial which is the constant1. Therefore, all the results in this paper are also valid for the case where F is a full row rankmatrix.We can define the concept of row reduced minors, and all the results in this paper can betranslated to similar results for FRP factorizations of multivariate polynomial matrices withoutfull column rank. We hope the results provided in the paper will motivate further research in thearea of factor prime factorizations. 15 cknowledgments This research was supported by the CAS Key Project QYZDJ-SSW-SYS022.
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