On Minor Left Prime Factorization Problem for Multivariate Polynomial Matrices
aa r X i v : . [ c s . S C ] O c t On Minor Left Prime Factorization Problem 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
A new necessary and su ffi cient condition for the existence of minor left prime factorizations ofmultivariate polynomial matrices without full row rank is presented. The key idea is to establisha relationship between a matrix and its full row rank submatrix. Based on the new result, wepropose an algorithm for factorizing matrices and have implemented it on the computer alge-bra system Maple. Two examples are given to illustrate the e ff ectiveness of the algorithm, andexperimental data shows that the algorithm is e ffi cient. Keywords:
Multivariate polynomial matrices, Polynomial matrix factorizations, Minor leftprime (MLP), Gr¨obner bases, Free modules
1. Introduction
Multivariate polynomial matrix factorization is one of the most important operations in mul-tidimensional systems, signal processing, and other related areas (Bose, 1982; Bose et al., 2003).The factorization problems of multivariate polynomial matrices have been extensively investi-gated and numerous algorithms have been developed to compute factorizations of multivariatepolynomial matrices. Since the factorization problems have been solved for univariate and bi-variate polynomial matrices (Morf et al., 1977; Guiver and Bose, 1982; Liu and Wang, 2013),we only consider the case where the number of variables is greater than or equal to three.Using three important concepts proposed by Youla and Gnavi (1979), there have been manypublications studying matrix factorizations. Lin (1999) first proposed the existence problem forzero prime factorizations of multivariate polynomial matrices. Charoenlarpnopparut and Bose(1999) first used Gr¨obner bases of modules to compute zero prime matrix factorizations ofmultivariate polynomial matrices. After that, Lin et al. (2008) introduced some applications ofGr¨obner bases in the broad field of signals and systems. Lin and Bose (2001) put forward thefamous Lin-Bose conjecture which was solved by Pommaret (2001); Wang and Feng (2004).Wang and Kwong (2005) focused on the existence problem for minor prime factorizations of ∗ Corresponding author
Email addresses: [email protected] (Dong Lu), [email protected] (Dingkang Wang), [email protected] (Fanghui Xiao) ultivariate polynomial matrices, and gave a necessary and su ffi cient condition. Wang (2007)designed an algorithm to compute factor prime factorizations of a class of multivariate polyno-mial matrices.In linear algebra as well as multidimensional systems, the factorization problems of mul-tivariate polynomial matrices without full row rank are important and deserve some attention(Youla and Gnavi, 1979; Lin, 1999). Up to now, few results have been achieved on factoriza-tions of multivariate polynomial matrices without full row rank (Lin and Bose, 2001; Guan et al.,2018, 2019). Therefore, this paper focuses on factorization problems of multivariate polynomialmatrices without full row rank. Motivated by the views in Lin and Bose (2001), we try to uselocal properties to study the existence for minor prime factorizations of multivariate polynomialmatrices without full row rank.The rest of the paper is organized as follows. In Section 2, we introduce some basic conceptsand present the problem that we are considering. We present in Section 3 a new necessary andsu ffi cient condition for the existence of minor left prime factorizations of multivariate polynomialmatrices without full row rank. In Section 4, we construct an algorithm based on the new result,and use two examples to illustrate the e ff ectiveness of the algorithm. A comparison with Guan’salgorithm and experimental data are presented in Section 5. We end with some concludingremarks in Section 6.
2. Preliminaries and Problem
Let n be the number of variables, and z be the n variables z , . . . , z n , where n ≥
3. Let k [ z ] bethe polynomial ring in z over k , where k is an algebraically closed field. Let k [ z ] l × m denote theset of l × m matrices with entries in k [ z ], where l ≤ m . Let F ∈ k [ z ] l × m , we use d i ( F ) to denotethe greatest common divisor of all the i × i minors of F , and I i ( F ) to represent the ideal generatedby all the i × i minors of F , where 1 ≤ i ≤ l and we stipulate that I ( F ) = k [ z ].We first recall the most important concept in the paper. Definition 1.
Let F ∈ k [ z ] l × m be of full row rank. Then F is said to be an minor left prime (MLP)matrix if all the l × l minors of F are relatively prime, that is, d l ( F ) is a nonzero constant. Let F ∈ k [ z ] m × l with m ≥ l , an MRP matrix can be similarly defined. We refer to Youla and Gnavi(1979) for more details about the concepts of zero left prime (ZLP) matrices and factor left prime(FLP) matrices.An MLP factorization of a multivariate polynomial matrix is formulated as follows. Definition 2.
Let F ∈ k [ z ] l × m with rank r, where ≤ r ≤ l. F is said to admit an MLPfactorization if F can be factorized as F = G F (1) such that G ∈ k [ z ] l × r , and F ∈ k [ z ] r × m is an MLP matrix. When Youla and Gnavi studied the structure of n -dimensional linear systems, they obtainedthe following MLP factorization lemma by using matrix theory. Lemma 3.
Let A = " A A A A ∈ k [ z ] l × m with rank r, where A ∈ k [ z ] r × r with det( A ) , , A ∈ k [ z ] r × ( m − r ) , A ∈ k [ z ] ( l − r ) × r , A ∈ k [ z ] ( l − r ) × ( m − r ) , and ≤ r ≤ l. If [ A A ] is an MLP atrix, then A A − is a multivariate polynomial matrix and A has an MLP factorization A = " I r × r A A − A A i . (2)In order to state conveniently the problem of this paper, we introduce the following conceptsand conclusions. Definition 4 (Matsumura and Reid (1989)) . Let K be a submodule of k [ z ] × m , and J be a nonzeroideal 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 { f , . . . , f s } ⊂ k [ z ] be a Gr¨obner basis of J , then K : J = K : h f , . . . , f s i = ( K : f ) ∩ · · · ∩ ( K : f s ) . (3)Here, we write K : h f i as K : f for any f ∈ k [ z ]. Definition 5 (Eisenbud (2013)) . Let K be a finitely generated k [ z ] -module, and k [ z ] × l φ −−→ k [ z ] × m → K → be a presentation of K , 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 linearmapping φ . Then the ideal Fitt j ( K ) = I m − j ( F ) is called the j-th Fitting ideal of K . Here, wemake the convention that Fitt j ( K ) = k [ z ] for j ≥ m, and that Fitt j ( K ) = for j < max { m − l , } . We remark that
Fitt j ( K ) only depends on K . Cox et al. (2005) showed that one obtains apresentation matrix of K by arranging a system of generators of the syzygy module of K asrows. Let H ∈ k [ z ] m × t be composed of a system of generators of K , then the syzygy module of K is defined as follows: Syz( K ) = { ~ u ∈ k [ z ] × m | ~ u H = ~ } .Let F ∈ k [ z ] l × m with rank r , and J be a nonzero ideal of k [ z ], where 1 ≤ r ≤ l . We use ρ ( F ) todenote the submodule of k [ z ] × m generated by the rows of F . Wang (2007) and Guan et al. (2018)proved that the rank of ρ ( F ) : J is r . Let F ∈ k [ z ] s × m be composed of a system of generators of ρ ( F ) : J , and F ∈ k [ z ] t × s be composed of a system of generators of Syz( F ), where s ≥ r and t ≥ s − r . Then, F is a presentation matrix of ρ ( F ) : J . Moreover, ρ ( F ) : J is a free moduleof rank r if and only if Fitt r ( ρ ( F )) generates k [ z ], that is, I s − r ( F ) = k [ z ]. We refer to Eisenbud(2013) for more details.Wang and Kwong (2005) proposed a necessary and su ffi cient condition for MLP factoriza-tions of multivariate polynomial matrices with full row rank. Lemma 6.
Let F ∈ k [ z ] l × m be of full row rank. Then the following are equivalent: F has an MLP factorization; ρ ( F ) : d l ( F ) is a free module of rank l. Guan et al. (2019) generalized Lemma 6 to the case of multivariate polynomial matrices with-out full row rank.
Lemma 7.
Let F ∈ k [ z ] l × m with rank r, where ≤ r ≤ l. Then the following are equivalent: F has an MLP factorization; ρ ( F ) : I r ( F ) is a free module of rank r. Remark 8.
Although Lemma 7 is di ff erent from Lemma 6 for the case of r = l, Guan et al. haveproved that ρ ( F ) : I l ( F ) = ρ ( F ) : d l ( F ) . Let a , . . . , a β ∈ k [ z ] be all the r × r minors of F , then I r ( F ) = h a , . . . , a β i , where β = (cid:16) lr (cid:17) · (cid:16) mr (cid:17) .From Equation (3) we have ρ ( F ) : I r ( F ) = ( ρ ( F ) : a ) ∩ · · · ∩ ( ρ ( F ) : a β ) . (4)When we verify whether ρ ( F ) : I r ( F ) is a free module of rank r , we need to do the followingcalculation. First, we compute a Gr¨obner basis { ¯ a , . . . , ¯ a γ } of I r ( F ), where γ ≤ β . Then, ρ ( F ) : I r ( F ) = ( ρ ( F ) : ¯ a ) ∩ · · · ∩ ( ρ ( F ) : ¯ a γ ) . (5)Second, we obtain a system G i of generators of ρ ( F ) : ¯ a i by computing a Gr¨obner basis of acorresponding module (we refer to Section 4 for more details), where i = , . . . , γ . Third, wecompute a Gr¨obner basis G of G ∩ · · · ∩ G γ . Finally, we compute a Gr¨obner basis of the r -thFitting ideal of the module generated by the elements in G .As we all know, the method of computing a Gr¨obner basis of the intersection of modules isto introduce new variables. Given that the complexity of Gr¨obner basis computations is heavilyinfluenced by the number of variables and the total degrees of polynomials (Mayr and Meyer,1982; M¨oller and Mora, 1984), it can be seen that the calculation amount of ρ ( F ) : I r ( F ) is verylarge. Therefore, we consider the following problem. Problem 9.
Is there a simpler condition that can replace ρ ( F ) : I r ( F ) in Lemma 7?
3. Main Result
Let F ∈ k [ z ] l × m with rank r , where 1 ≤ r ≤ l . We use Lemma 3 to establish a relationshipbetween F and an arbitrary full row rank submatrix of F , and then solve Problem 9. Theorem 10.
Let F ∈ k [ z ] l × m with rank r, and F ∈ k [ z ] r × m be an arbitrary full row ranksubmatrix of F , where ≤ r ≤ l. Then the following are equivalent: F has an MLP factorization; ρ ( F ) : d r ( F ) is a free module of rank r.Proof. →
2. Suppose F has an MLP factorization. Then there exist G ∈ k [ z ] l × r and F ∈ k [ z ] r × m such that F = G F with F being an MLP matrix. Without loss of generality, we assumethat the first r rows of F are k [ z ]-linearly independent. Let F ∈ k [ z ] r × m be composed of the first r rows of F , then F = " F C = " G G F , (6)where G ∈ k [ z ] r × r is the first r rows of G . From Equation (6) we have F = G F . (7)According to Lemma 6, ρ ( F ) : d r ( F ) is a free module of rank r .4 →
1. Assume that ρ ( F ) : d r ( F ) is a free module of rank r . Using Lemma 6, there exist G ∈ k [ z ] r × r and F ∈ k [ z ] r × m such that F = G F with F being an MLP matrix. Since F is an arbitrary r × m submatrix of F , there exists an elementary transformation matrix U ∈ k l × l such that F is the first r rows of ¯ F , where ¯ F = UF . Let ¯ F = [ F T1 C T ] T , where C ∈ k [ z ] ( l − r ) × m isthe last ( l − r ) rows of ¯ F . Then,¯ F = UF = " F C = " G F C = " G r × ( l − r ) ( l − r ) × r I ( l − r ) × ( l − r ) F C . (8)Because F ∈ k [ z ] r × m is a full row rank matrix, there exists another elementary transformationmatrix V ∈ k m × m such that the first r columns of ¯ F are k [ z ]-linearly independent, where ¯ F = F V . It follows from det( V ) = F V − = F . According to the Binet-Cauchy formula,we obtain d r ( ¯ F ) | d r ( F ). This implies that d r ( ¯ F ) is a nonzero constant. Therefore, ¯ F is anMLP matrix. Suppose that " F C V = " A A A A , (9)where A ∈ k [ z ] r × r , A ∈ k [ z ] r × ( m − r ) , A ∈ k [ z ] ( l − r ) × r , and A ∈ k [ z ] ( l − r ) × ( m − r ) . Then,det( A ) , A A ] is an MLP matrix. By Lemma 3, we get " F C V = " I r × r A A − A A i = " I r × r A A − ¯ F . (10)Combining Equation (8) and Equation (10), we have UFV = " G r × ( l − r ) ( l − r ) × r I ( l − r ) × ( l − r ) I r × r A A − ¯ F = " G A A − ¯ F . (11)As U and V are two elementary transformation matrices, from Equation (11) we can derive F = U − " G A A − ¯ F V − = U − " G A A − F . (12)Let G = U − " G A A − and F = F , then F = G F . Thus, F has an MLP factorization, andthe proof is completed. Remark 11.
Theorem 10 is the same as Lemma 6 for the case of r = l. According to the proof process of su ffi ciency in Theorem 10, we can propose a new con-structive algorithm to compute an MLP factorization of F . We will introduce the new algorithmin detail in the following section.
4. Algorithm and Examples
Let F ∈ k [ z ] l × m with rank r , and F ∈ k [ z ] r × m be an arbitrary full row rank submatrix of F , where 1 ≤ r ≤ l . Suppose ρ ( F ) : d r ( F ) is a free module of rank r , then F has an MLPfactorization. Now, we need to design an algorithm to compute G ∈ k [ z ] l × r and F ∈ k [ z ] r × m such that F = G F with F being an MLP matrix.5omputing free bases of free modules is a crucial step in the process of matrix factoriza-tions. Fabia´nska and Quadrat (2007) first designed a Maple package, which is called QUILLEN-SUSLIN, to compute free bases of free modules. Based on this fact, we will implement ouralgorithm on Maple.We have two problems to solve. The first one is how to compute a system of generators of ρ ( F ) : d r ( F ), and another one is how to compute G ∈ k [ z ] r × r such that F = G F , where F is composed of a free basis of ρ ( F ) : d r ( F ). We can use the commands “quotient” and“lift” on the computer algebra system Singular (Decker et al., 2016) to solve the two problems.However, we need to solve these problems on Maple.Wang and Kwong (2005) proved that there are one to one correspondences between the twomodules: ρ ( F ) : d r ( F ) and Syz([ F T1 − d r ( F ) · I m × m ] T ). That is, we compute a Gr¨obner basis { [ ~ g , ~ f ] , . . . , [ ~ g s , ~ f s ] } of Syz([ F T1 − d r ( F ) · I m × m ] T ), then { ~ f , . . . , ~ f s } is a system of generators of ρ ( F ) : d r ( F ), where [ ~ g i , ~ f i ] ∈ k [ z ] × ( r + m ) and i = , . . . , s .Now, we solve the second problem. Let F be composed of { ~ f , . . . , ~ f r } and F be composedof { ~ h , . . . , ~ h r } , where ~ f i , ~ h j ∈ k [ z ] × m and 1 ≤ i , j ≤ r . It follows from ρ ( F ) ⊂ ρ ( F ) that ~ f i ∈ h ~ h , . . . , ~ h r i for each i . According to the division algorithm in k [ z ] × m (Cox et al., 2005), weuse { ~ h , . . . , ~ h r } to reduce ~ f i and obtain the following equation: ~ f i = a i ~ h + · · · + a ir ~ h r + ~ v i , i = , . . . , r , (13)where a i j ∈ k [ z ] and ~ v i ∈ k [ z ] × m . However, ~ v i may be a nonzero vector since { ~ h , . . . , ~ h r } is nota Gr¨obner basis. Hence, we first need to compute a Gr¨obner basis { ~ g , . . . , ~ g s } of h ~ h , . . . , ~ h r i ,where s ≥ r . In the calculation process, we record the relationship between { ~ h , . . . , ~ h r } and { ~ g , . . . , ~ g s } . That is, ~ g i = p i ~ h + · · · + p ir ~ h r , i = , . . . , s . (14)Then, we use { ~ g , . . . , ~ g s } to reduce ~ f i and get ~ f i = q i ~ g + · · · + q is ~ g s , i = , . . . , r . (15)Let P = p · · · p r ... ... ... p s · · · p sr and Q = q · · · q s ... ... ... q r · · · q rs . Combining Equation (14) and Equation (15), wehave F = G F = ( QP ) F . (16)Lu et al. (2020) designed a Maple package, which is called poly-matrix-equation, for solvingmultivariate polynomial matrix Diophantine equations. We can use this package to implementthe above calculation process.Now, we can propose a new constructive algorithm to compute MLP factorizations of poly-nomial matrices without full row rank.From Algorithm 1 we have ρ ( F ′ ) = ρ ( F ) : d r ( F ) in step 6 and I s − r ( H ) = Fitt r ( ρ ( F ′ )) instep 7. Moreover, G , { } in step 8 implies that ρ ( F ) : d r ( F ) is not a free module of rank r . If s = r in step 6, then F ′ is a full row rank matrix. It follows that ρ ( F ) : d r ( F ) is a free moduleof rank r and the rows of F ′ constitute a free basis of ρ ( F ) : d r ( F ). In this case, we do not needto compute a Gr¨obner basis of Syz( F ′ ) and perform the calculation from step 11.We use the two examples in Guan et al. (2019) to illustrate the calculation process of Algo-rithm 1. 6 lgorithm 1: MLP factorizations
Input : F ∈ k [ z ] l × m . Output: an MLP factorization of F . begin compute the rank r of F ; perform elementary row transformations on F , such that the first r rows of ¯ F are k [ z ]-linearly independent, where ¯ F = UF and U ∈ k l × l is an elementarytransformation matrix; compute d r ( F ), where F is composed of the first r rows of ¯ F ; compute a Gr¨obner basis { [ ~ g , ~ f ] , . . . , [ ~ g s , ~ f s ] } of Syz([ F T1 − d r ( F ) · I m × m ] T ); compute a Gr¨obner basis { ~ h , . . . , ~ h t } of Syz( F ′ ) and use it to constitute H ∈ k [ z ] t × s ,where F ′ ∈ k [ z ] s × m is composed of { ~ f , . . . , ~ f s } ; compute a Gr¨obner basis G of I s − r ( H ); if G , { } then return F has no MLP factorizations. use the QUILLENSUSLIN package to compute a free basis of ρ ( F ′ ) and use it tomake up F ∈ k [ z ] r × m ; use the poly-matrix-equation package to compute G ∈ k [ z ] r × r such that F = G F ; perform elementary column transformations on F , such that the first r columns of¯ F are k [ z ]-linearly independent, where ¯ F = F V and V ∈ k m × m is anelementary transformation matrix; compute CV , where C ∈ k [ z ] ( l − r ) × m is the last ( l − r ) rows of ¯ F ; compute A A − , where A is composed of the first r columns of ¯ F , and A iscomposed of the first r columns of CV ; return ( U − " G A A − , F ). 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 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 the first 2 rows of F are C [ z , z , z ]-linearlyindependent. Let F ∈ C [ z , z , z ] × be composed of the first 2 rows of F , then d ( F ) = z z − z . We compute a Gr¨obner basis of Syz([ F T1 − d ( F ) · I × ] T ) and obtain { [0 , z , z z , , z + , [ z − , , z z + z , , } . Now, we get a system of generators of ρ ( F ) : d ( F ) as follows { [ z z , , z + , [ z z + z , , } . F ′ = " z z z + z z + z . Since rank( F ′ ) = F ′ is a full row rank matrix. Then, ρ ( F ) : d ( F ) is a free module ofrank 2, and the rows of F ′ constitute a free basis of ρ ( F ) : d ( F ). Let F = F ′ , we use thepoly-matrix-equation package to compute G ∈ C [ z , z , z ] × such that F = G F , andobtain G = " z z − . Note that the first 2 columns of F are C [ z , z , z ]-linearly independent. Let A = " z z z z + z and A = h z z z − z z + z z − z z z − z i , then A A − = h z z z z − z i . Therefore, F has an MLP factorization: F = " G A A − F = z z − z z z z − z " z z z + z z + z . Example 13.
Let F = 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 the first 2 rows of F are C [ z , z , z ]-linearlyindependent. Let F ∈ C [ z , z , z ] × be composed of the first 2 rows of F , then d ( F ) = z + F T1 − d ( F ) · I × ] T ) and obtain a system of generators of ρ ( F ) : d ( F ) as follows { [ 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 , then a Gr¨obner basis of Syz( F ′ ) is { [ − z , z − , − z − } . Let H = [ − z z − − z − Fitt ( ρ ( F ′ )) = I ( H ) , C [ z , z , z ] . This implies that ρ ( F ) : d ( F ) is not a free module of rank 2. Then, F has no MLP factoriza-tions. 8 . Comparative Performance The above two examples show that Algorithm 1 is simpler than the algorithm, which is calledGuan’s algorithm, proposed by Guan et al. (2019). To illustrate the advantages of our algorithm,we first compare the main di ff erences between the two algorithms. Table 1: The comparison of two MLP factorization algorithms
Main step Guan’s algorithm Algorithm 11 ρ ( F ) : I r ( F ) ρ ( F ) : d r ( F )2 F = G F F = G F and A A − The symbols in the above table are the same as those in Lemma 7 and Theorem 10. FromTable 1, we can get the following preliminary conclusions: first, the calculation of the main step1 of Algorithm 1 is faster than that of Guan’s algorithm in almost all cases; second, althoughin Algorithm 1 we need to compute A A − additionally, the scale of equation F = G F issmaller than that of equation F = G F .Next, we will show from the specific experimental data that Algorithm 1 is more e ffi cientthan Guan’s algorithm. The two algorithms have been implemented by us on the computer al-gebra system Maple. The implementations of the two algorithms have been tried on a numberof examples including the two examples in Section 4. Please see the Appendix A for all ex-amples. For interested readers, more comparative examples can be generated by the codes at: . Table 2: Comparative performance of MLP factorization algorithms
Example Guan’s algorithm t (sec) Algorithm 1 t (sec) Time comparison t / t F F F F F F F F
6. Concluding Remarks
We have given a new necessary and su ffi cient condition for the existence of MLP factoriza-tions of multivariate polynomial matrices in this paper. All cases with matrices being full rowrank and non-full row rank are considered. Based on the new result, a constructive algorithmfor computing MLP factorizations has been proposed. We have implemented Algorithm 1 andGuan’s algorithm on Maple, and the experimental data in Table 2 suggests that Algorithm 1 issuperior in practice in comparison with Guan’s algorithm. This is due to the fact that we can9etermine whether F has an MLP factorization through less calculations and requires less timeto calculate G . Acknowledgments
This research was supported in part by the CAS Key Project QYZDJ-SSW-SYS022.
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F = z z + z − z − z z + z + z − z z + z z + z + z − . F ∈ C [ z , z , z ] × is as follows, and it has no MLP factorizations. 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 . F ∈ C [ z , z , z ] × is as follows, and it has an MLP factorization. F = z z z z z z + z z z z z z z z z . F ∈ C [ z , z , z ] × is as follows, and it has an MLP factorization. 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 . F ∈ C [ z , z , z ] × is as follows, and it has an MLP factorization. F [1 , = z − z , F [1 , = − z z + z − z , F [1 , = z z − z − z , F [2 , = 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 − 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 z z + z z + z z − z z + z z − z − z , F [3 , = z z − z z − z z + z z + z z − z , F [3 , = − z z − z z − z z − z z − z z + z + z + z , F [3 , = z z z − z z − z z z − z z + z z + z + z z − z . F ∈ C [ z , z , z ] × is as follows, and it has an MLP factorization. F [1 , = z z + z z + 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 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 − z z + z − z z − z − z + z − , F [2 , = z z − z + z z − z z − z + , F [2 , = 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 + , F [3 , = z z z + z z + z z − z z z + z − z z − z z − z , F [3 , = − z z + z z z − z z − z z + z z + z z − z z − z + z − z , F [3 , = z z z − z z z + z z − z z − z z − z z − z z − z − z − z . F ∈ C [ z , z , z ] × is as follows, and it has an MLP factorization. F [1 , = 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 z + z , F [1 , = z z z − z z z + z z − z z + z − z , F [1 , = 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 , F [2 , = − z z + z z − z z z − z z − z z − z , F [2 , = − z + z , F [2 , = − z z , F [2 , = − z z − z + z , F [3 , = − 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 [3 , = − z z z + z z − z z + z z − z z − z z + z + z , F [3 , = − z z − z z − z − , F [3 , = − z z z − z z + z z − z z z − z z + z z − z z − z z − z z + z + z − z . F ∈ C [ z , z , z ] × is as follows, and it has an MLP factorization. F [1 , = 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 [1 , = 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 − 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 , F [2 , = − 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 + z z + z z z − z z − z + z − , F [2 , = − z z z + z z + z z − z z − z , F [3 , = 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 [3 , = 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 − z z + z + z z − z − z − , F [3 , = z z + z z z + z z + z z − z z z − z z − z + z z − z z − z ..