# Minor Prime Factorization forn-D Polynomial Matrices over Arbitrary Coefficient Field

Authors: Jinwang Liu; Dongmei Li; Licui Zheng
Journal: Complexity
 (2018)

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## Abstract

In this paper, we investigate two classes of multivariate (n-D) polynomial matrices whose coefficient field is arbitrary and the greatest common divisor of maximal order minors satisfy certain condition. Two tractable criterions are presented for the existence of minor prime factorization, which can be realized by programming and complexity computations. On the theory and application, we shall obtain some new and interesting results, giving some constructive computational methods for carrying out the minor prime factorization.


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## Body

## 1. Introduction

In the polynomial approach, pioneered by Rosenbrock [1], matrices over Rs, s≡d/dt are used to represent linear systems of ordinary differential equations. For more general linear functional systems, e.g., partial differential systems or delay-differential systems, the resulting system matrices are multivariate. The problem of multivariate (n-D) polynomial matrix has attracted much attention over the past decades because of its diverse applications in n-D systems, network theory, Wiener-Hopf equations, controls, and signal processing ([2–8]); its factorizations have attracted much attention over the past decades because of their wide applications ([9–25]). Lin-Bose conjecture is one of the important problems of n-D polynomial matrix factorizations, which has been proved in [15–20]. It gave a sufficient condition for a matrix to have zero prime factorization or minor prime factorization. After that, Lin et al. proposed a tractable criterion for the existence of minor prime factorizations for a class of n-D polynomial matrices whose greatest common divisor of maximal order minors is z1−fz2,⋯,zn. They have presented a constructive method for carrying out the minor prime factorization when it exists [14], where the coefficient field is the complex number field. In this paper, we investigate the n-D polynomial matrices whose coefficient field is arbitrary and minor prime factorizations for two classes of n-D polynomial matrices whose the greatest common divisor of maximal order minors is z1−fz2,⋯,zn or ∏k=1pz1−z1k. In this case, because the coefficient field is an arbitrary field, the many previous results (lemmas) cannot be used directly. By giving some new key lemmas, two tractable criterions for the existence of minor prime factorization are obtained. These criterions can be judged by using “programming of reduced Gröbner basis.” Furthermore, we give constructive computational methods for carrying out the minor prime factorization.


## 2. Notations and Preliminaries

For an arbitrary fieldK, let Kz1,z2,⋯,zn denote the set of polynomial ring in n variables z1,z2,…,zn with coefficients in K and K¯ denote an algebraic closed field of K. For the convenience, Kz1,z2,⋯,zn is denoted simply by Kz; when K=C is the complex number field, Cz1,z2,⋯,zn is denoted simply by Cz. Let Kl×mz denote the set of l×m matrices with entries in Kz and 0r×t denote the r×t zero matrices. Throughout the paper, the argument z is omitted whenever its omission does not cause confusion.Definition 2.1 (see [21]).
LetFz∈Kl×mz be of normal rank r and a1z,a2z,…,aβz denote all the r×r minors of the matrix Fz. Extracting the greatest common divisor (g.c.d.), denoted by dF or d, of a1z,a2z,…,aβz gives
(1)aiz=dbiz,i=1,2,…,β.
Then,b1z,b2z,…,bβz are called the reduced minors of Fz.Definition 2.2.
LetFz∈Kl×mz be of normal full rank, then Fz is said to be zero left prime (zero right prime) if the l×lm×m minors of Fz generate the unit ideal Kz, which is simplified as that Fz is ZLP (ZRP).
First, the following two lemmas and a well-known result (Theorem 1 in [14]) are required.Lemma 2.1.
Letgz∈Kz and fz∈Kz2,⋯,zn. Supposing that gf,z2,⋯,zn=0, then z1−fz2,⋯,zn is a divisor of gz.
The proof is simple and omitted.Lemma 2.2.
Letf1z,f2z,…,fsz∈Kz be polynomials, then they have no common zeros in K¯n (are zero coprime) if and only if there are u1z,u2z,…,usz∈Kz such that
(2)u1zf1z+u2zf2z+⋯+uszfsz=1,i.e., f1z,f2z,⋯,fsz is a ZLP row vector in Knz, or f1z,…,fsz generate the unit ideal Kz.Proof 1.
Necessity. It is clear that fiz∈K¯z,i=1,2,…,s. Let IK and IK¯ denote the ideal generated by f1z,f2z,…,fsz in Kz and in K¯z, respectively. From the assumption and Hilbert Zero Theorem, IK¯=K¯z. Then the unit element 1 (in K¯) may be represented by a Gröbner basis G of IK¯, i.e., 1 may be reduced to 0 by G. According to computational methods of G, we have that G⊂Kz, then 1 can be represented by G in Kz, i.e., there are u1z,u2z,…,usz∈Kz such that
(3)u1zf1z+u2zf2z+⋯+uszfsz=1.
Sufficiency. We assume that f1z,f2z,…,fsz have common zeros in K¯n and z0=z11,z21,⋯,zn1 is one of them. From (2), we have
(4)u1z0f1z0+u2z0f2z0+⋯+usz0fsz0=0≠1.
This is a contradiction. Sof1z,f2z,…,fsz have no common zeros in K¯n.Theorem 2.1 (see [14]).
LetFz∈Cm×lz, dF=z1−fz2,⋯,zn. Then a necessary and sufficient condition for Fz to admit a minor prime factorization Fz=F0zG0z with F0z∈Km×lz,G0z∈Kl×lz, and detG0z=dF is that there exists a full rank l−1×l submatrix F1z2,⋯,zn≜F1fz2,⋯,zn,z2,⋯,zn of Ffz2,⋯,zn,z2,⋯,zn such that the reduced minors of F1z2,⋯,zn are zero coprime.


## 3. Main Results

For the complex number fieldC and an n-D polynomial matrix Fz1,z2,⋯,zn∈Cz, from Lemma 2.2, we see that the reduced minors of Fz1,z2,⋯,zn are zero coprime if and only if the reduced minors of Fz1,z2,⋯,zn generate the unit ideal Cz. In the following, we investigate the case in which Fz1,z2,⋯,zn∈Kz, where K is an arbitrary field (may be a finite field); we shall extend the range of applicability of Theorem 2.1 and present our main results. First, several key lemmas are presented.Lemma 3.1.
LetQ∈Kl−1×lz be of normal full rank, if the reduced minors of Q generate Kz, then there is a ZLP matrix w∈Kl×1z such that Q⋅w=0l−1×1.Proof 2.
Since the reduced minors ofQ generate the unit ideal Kz, by Lin-Bose Theorem (Theorem 3.3 in [18]), Q can be factorized as Q=Q1B1 such that Q1∈Rl−1×l−1 is of normal full rank, and B1∈Kl−1×l is ZRP. By Quillen-Suslin Theorem [26], there is a unimodular U1∈Kl×lz such that B1 is its first l−1 rows, i.e., U1=B1TV1TT, where V1∈K1×l. Let U1−1=V2w, where V2∈Rl×l−1, w∈Rl×1, then B1w=0l−1×1. Obviously, w is ZLP. We have that Q⋅w=Q1⋅B1⋅w=Q1⋅0l−1×l=0l−1×1.Lemma 3.2.
LetFz∈Km×lzl≤m and dF=∏s=1qz1−fsz2,⋯,zn, where fi−fj≠0 for i≠j. Then rank Ffpz2,⋯,zn,z2,⋯,zn is l−1, where p=1,2,…,q.Proof 3.
LetFz=fijm×l, for p∈1,2,⋯,q, we may write fij as
(5)fij=qijz1−fpz2,⋯,zn+bij,where qij∈Kz, bij∈Kz2,⋯,zn,i=1,2,…,m,j=1,2,…,l.
SettingA≜aijm×l≜qijz1−fpz2,⋯,znm×l, B≜bijm×l, we have that F=A+B. Since dF=∏s=1qz1−fsz2,⋯,zn, then there is an l×l minor Δ of Fz such that z1−fpz2,⋯,zn is a simple divisor of Δ. Let Fj1,j2,⋯,jl denote an l×l submatrix consisting of the j1,j2,…,jl rows of F such that z1−fpz2,⋯,zn is a simple divisor of detFj1,j2,⋯,jl. Then
(6)Fj1,j2,⋯,jl=aj11+bj11aj12+bj12⋯aj1l+bj1laj21+bj21aj22+bj22⋯aj2l+bj2l⋯⋯⋯⋯ajl1+bjl1ajl2+bjl2⋯ajll+bjll.
Sincez1−fpz2,⋯,zn∣aij, where i=1,2,…,m, j=1,2,…,l, bij is independent of z1, and z1−fpz2,⋯,zn is a simple divisor of detFj1,j2,⋯,jl, by the properties of determinant, some k1≤k≤l exists such that
(7)bj11⋯bj1k−1aj1kbj1k+1⋯bj1lbj21⋯bj2k−1aj2kbj2k+1⋯bj2l⋯⋯⋯⋯⋯⋯⋯bjl1⋯bjlk−1ajlkbjlk+1⋯bjll≢0.
From (7) and Laplace Theorem, there is a permutation i1i2⋯il−1 of j1j2⋯jl such that
(8)Δk≜bi11⋯bi1k−1bi1k+1⋯bi1lbi21⋯bi2k−1bi2k+1⋯bi2l⋯⋯⋯⋯⋯⋯bil−11⋯bil−1k−1bil−1k+1⋯bil−11≢0.
It is obvious thatFfpz2,⋯,zn,z2,⋯,zn=B, then Δk is an l−1×l−1 minor of Ffpz2,⋯,zn,z2,⋯,zn. Since dFfpz2,⋯,zn,z2,⋯,zn=0, combined with (8), we have that rank Ffpz2,⋯,zn,z2,⋯,zn is l−1.Lemma 3.3.
LetFz∈Kl×lz and detF=∏s=1qz1−fsz2,⋯,zn, where 0≠fi−fj∈K for i≠j. Then rank Ffpz2,⋯,zn,z2,⋯,zn=l−1 for every z2,⋯,zn∈K¯n−1, and rank Ffpz2,⋯,zn,z2,⋯,zn=l−1; furthermore, the l−1×l−1 minors of Ffpz2,⋯,zn,z2,⋯,zn generate the unit ideal Kz, where p=1,2,…,q.Proof 4.
For arbitrary but fixedz2=z20,…,zn=zn0, where z20,⋯,zn0∈K¯n−1, let z1p=fpz20,⋯,zn0, then z11,z12,…,z1q are distinguished. Assuming that Fz1,z20,⋯,zn0=fijl×l, for p∈1,2,⋯,q, we may write fij as
(9)fij=qijz1−z1p+bij,where qij∈K¯z1, bij∈K¯,i,j=1,2,…,l.
SettingA≜aijl×l≜qijz1−z1pl×l, B≜bijl×l, then we have that Fz1,z20,⋯,zn0=A+B,
(10)detFz1,z20,⋯,zn0=a11+b11a12+b12⋯a1l+b1la21+b21a22+b22⋯a2l+b2l⋯⋯⋯⋯al1+bl1al2+bl2⋯all+bll.
SincedetFz1,z20,⋯,zn0=∏s=1qz−z1s,z1−z1p∣aij, where i,j=1,2,…,l, and bij∈K¯, z1−z1p is a simple divisor of detFz1,z20,⋯,zn0, by the properties of determinant, there is some k1≤k≤l such that
(11)b11⋯b1k−1a1kb1k+1⋯b1lb21⋯b2k−1a2kb2k+1⋯b2l⋯⋯⋯⋯⋯⋯⋯bl1⋯blk−1alkblk+1⋯bll≢0.
From (11) and Laplace Theorem, there is a permutation i1i2⋯il−1 of 12⋯l such that
(12)Δk≜bi11⋯bi1k−1bi1k+1⋯bi1lbi21⋯bi2k−1bi2k+1⋯bi2l⋯⋯⋯⋯⋯⋯bil−11⋯bil−1k−1bil−1k+1⋯bil−11≢0.
It is obvious thatFfpz20,⋯,zn0,z20,⋯,zn0=B, then Δk is an l−1×l−1 minor of Ffpz20,⋯,zn0,z20,⋯,zn0. Since detFfpz20,⋯,zn0,z20,⋯,zn0=0, combined with (12), we have that rank Ffpz2,⋯,zn,z2,⋯,zn is l−1 for every z2,⋯,zn∈K¯n−1, i.e., the l−1×l−1 minor of Ffpz2,⋯,zn,z2,⋯,zn have no common zeros in K¯n−1. Noting that detFfpz2,⋯,zn,z2,⋯,zn=0, then rank Ffpz2,⋯,zn,z2,⋯,zn=l−1. Furthermore, by Lemma 2.2, we obtain that the l−1×l−1 minors of Ffpz2,⋯,zn,z2,⋯,zn generate the unit ideal Kz.Corollary 3.1.
LetFz∈Kl×lz with detFz=∏k=1qz1−z1k, where z11,z12,…,z1q∈K are distinguished. Then rank Fz1k,z2,⋯,zn=l, the l−1×l−1 minors of Fz1k,z2,⋯,zn generate the unit ideal Kz, where k=1,2,…,q.Lemma 3.4.
LetGz∈Kl×lz, detGz=∏s=1qz1−fsz2,⋯,zn, where 0≠fi−fj∈K for i≠j. Then, for p∈1,2,⋯,q,
(13)Gfpz2,⋯,zn,z2,⋯,zn=G¯z⋅diag0,1,⋯,1⋅Vp,where G¯z∈Kl×lz2,⋯,zn, Vp∈Kl×lz2,⋯,zn is an invertible matrix.Proof 5.
SincedetGz=∏s=1qz1−fsz2,⋯,zn, for p∈1,2,⋯,q, by Lemma 3.3, rank Gfpz2,⋯,zn,z2,⋯,zn=l−1 and the l−1×l−1 minors of Gfpz2,⋯,zn,z2,⋯,zn generate Kz. By Lemma 3.1, there is a ZLP vector wp∈Kl×1z2,⋯,zn such that G⋅Ffpz2,⋯,zn,z2,⋯,zn⋅wpz2,⋯,zn=0,⋯,0T. According to Quillen-Suslin Theorem [26], a unimodular matrix U1∈Kl×lz2,⋯,zn can be constructed such that wpz2,⋯,zn is its first column, then
(14)Gfpz2,⋯,zn,z2,⋯,zn⋅U1=G¯z⋅diag0,1,⋯,1,where G¯z∈Kl×lz2,⋯,zn. Setting Vp=U1−1, then the desired results are obtained.Lemma 3.5.
LetFz∈Kl−1×lz, if Fz is of normal full rank and Fz has the following factorization:
(15)Fz=Az⋅diag0,1,⋯,1⋅Vz,where Az∈Kl−1×lz, Vz∈Kl×lz is a unimodular matrix. Then the reduced minors of Fz generate the unit ideal Kz.Proof 6.
We partitionAz, Vz as the following form, respectively,
(16)Az=A1zA2z,Vz=V1TzV2TzT,where A1z∈Kl−1×1z, A2z∈Kl−1×l−1z, V1z∈K1×lz, V2z∈Kl−1×lz. Then V2z is ZLP, i.e., the reduced minors of V2z generate the unit ideal Kz, and
(17)Fz=0A2z⋅V1TzV2TzT=A2z⋅V2z.
So the reduced minors ofFz are the same as that of V2z, which generate the unit ideal Kz.Theorem 3.1.
LetK be an arbitrary field, Fz∈Km×lz be of normal full rank l≤m, dF=z1−fz2,⋯,zn≜dz. Then a necessary and sufficient condition for Fz to admit a minor prime factorization Fz=F0zG0z with F0z∈Km×lz,G0z∈Kl×lz, and detG0z=dz is that there exists a normal full rank l−1×l submatrix F1z2,⋯,zn≜F1fz2,⋯,zn,z2,⋯,zn of Ffz2,⋯,zn,z2,⋯,zn such that the reduced minors of F1z2,⋯,zn generate the unit ideal Kz.Proof 7.
Necessity. Supposing that Fz admits a minor prime factorization
(18)Fz=F0zG0z,with F0z∈Km×lz, G0z∈Kl×lz, and detG0z=z1−fz2,⋯,zn=dz. From Lemma 3.4, G0fz2,⋯,zn,z2,⋯,zn=G0¯z⋅diag0,1,⋯,1⋅V, where V∈Kz2,⋯,zn is unimodular. Combined with (18), we have
(19)Ffz2,⋯,zn,z2,⋯,zn=F0fz2,⋯,zn,z2,⋯,zn⋅G0fz2,⋯,zn,z2,⋯,zn=F0fz2,⋯,zn,z2,⋯,zn⋅G0¯z⋅diag0,1,⋯,1⋅V.
By Lemma3.2, we see that the normal rank of Ffz2,⋯,zn,z2,⋯,zn is l−1. Hence, from (19), there exists a normal full rank l−1×l submatrix F1z2,⋯,zn≜F1fz2,⋯,zn,z2,⋯,zn of Ffz2,⋯,zn,z2,⋯,zn such that
(20)F1z2,⋯,zn=Az⋅diag0,1,⋯,1⋅V,where Az is a l−1×l submatrix of F0fz2,⋯,zn,z2,⋯,zn⋅G0¯z. By Lemma 3.5, the reduced minors of F1z2,⋯,zn generate the unit ideal Kz.
Sufficieny. Assuming that there exists a normal full rank l−1×l submatrix F1z2,⋯,zn≜F1fz2,⋯,zn,z2,⋯,zn of Ffz2,⋯,zn,z2,⋯,zn such that the reduced minors of F1z2,⋯,zn generate the unit ideal Kz. According to Lemma 3.1, a ZRP vector ω∈Kl×1z2,⋯,zn can be constructed such that
(21)F1z2,⋯,zn⋅ωz2,⋯,zn=0,⋯,0T.
SinceFfz2,⋯,zn,z2,⋯,zn is of normal rank l−1, by (21) and the fact that F1z2,⋯,zn is a normal full rank submatrix of Ffz2,⋯,zn,z2,⋯,zn, we have
(22)Ffz2,⋯,zn,z2,⋯,zn⋅ωz2,⋯,zn=0,⋯,0T.
By Quillen-Suslin Theorem [26], a unimodular square matrix V∈Kl×lz2,⋯,zn can be constructed such that ωz2,⋯,zn is its first column; combined with Lemma 2.1, we have
(23)Fz⋅Vz2,⋯,zn=F0z⋅diagz1−fz2,⋯,zn,1,⋯,1,where F0z∈Km×lz. It follows that
(24)Fz=F0zG0z,where G0z=diagz1−fz2,⋯,zn,1,⋯,1⋅V−1∈Kl×lz with detG0z=z1−fz2,⋯,zn=dF.Remark 3.1.
WhenK=C is the complex number field, according to Hilbert Zero Theorem, we see that the condition in Theorem 3.1 is the same as that of Theorem 2.1, and Theorem 3.1 is a generalization of Theorem 2.1.According to Theorem3.1, in order to check that Fz satisfies the conditions of Theorem 3.1, we first find all the l−1×l submatrices of Ffz2,⋯,zn,z2,⋯,zn, denoted by F1,F2,…,Fs, where s=Cml−1=mm−1⋯m−l/l−1!. For every Fi1≤i≤s, we compute the l−1×l−1 minors of Fi, saying as ai1,ai2,…,ail. Let the ideal Ii=ai1,ai2,⋯,ail, we compute the reduced Gröbner basis Gi of Ii. If Gi contains only a polynomial diz, by the following result (Theorem 3.2), the reduced minors of Fi generate the unit ideal Kz.Theorem 3.2.
Letdiz be the greatest common divisor of ai1,ai2,…,ail, i.e.,
(25)aij=bjdiz,j=1,2,…,l,and Gi be the reduced Gröbner basis of the ideal Ii=ai1,ai2,⋯,ail. Then Gi contains only a polynomial ciz if and only if b1,b2,…,bl generate the unit Kz.Proof 8.
Necessity. Assuming that Gi=ciz, it is obvious that ciz∣aij, where j=1,2,…,l. Then ciz∣diz. Setting diz=piz⋅ciz, since ciz∈Ii, we have that there are u1,u2,…,ul such that
(26)ciz=u1ai1+u2ai2+⋯+ulail=u1b1diz+u2b2diz+⋯+ulbldiz,diz=pizciz=piu1b1diz+piu2b2diz+⋯+piulbldiz.
So1=piu1b1+piu2b2+⋯+piulbl. Thus, b1,b2,…,bl generate the unit ideal Kz.
Sufficiency. Assuming that b1,b2,…,bl generate the unit ideal Kz, then the reduced Gröbner basis of the ideal b1,b2,⋯,bl is 1. Furthermore, the reduced Gröbner basis Gi of Ii=b1diz,b2diz,⋯,bldiz is diz.
The reduced Gröbner basis algorithm of an ideal can be found in [16].Remark 3.2.
From the discussion above, we see that judging whetherFz satisfies the conditions of Theorem 3.1 or not, we only need to compute the reduced Gröbner basis of Ii, where i=1,2,…,l.Lin et al. studied the matrix factorization for a class of matrix whose g.c.d. of maximal order minors has divisor∏k=1pz1−z1k ([13]); in the following, we study the minor prime factorization for this class of matrices and propose a tractable criterion.Theorem 3.3.
LetK be an arbitrary field, Fz∈Km×ll≤m be of normal full rank, dF=∏k=1pz1−z1k, where z11,z12,…,z1p∈K are distinguished. Then Fz admits a minor prime factorization Fz=F0zG0z with F0z∈Km×lz, G0z∈Kl×lz, and detG0z=dF if and only if there are normal full rank l−1×l submatrices Fkz2,⋯,zn≜Fkz1k,z2,⋯,zn of Fz1k,z2,⋯,zn such that the reduced minors of Fkz2,⋯,zn generate the unit ideal Kz, where k=1,2,…,p.Proof 9.
Necessity. Supposing that Fz admits a minor prime factorization as follows:
(27)Fz=F0zG0z,where F0z∈Km×lz, G0z∈Kl×lz, and detG0z=dF. For arbitrary but fixed k1≤k≤p, from (27) and Lemma 3.4, we have
(28)Fz1k,z2,⋯,zn=F0z1k,z2,⋯,zn⋅G0z1k,z2,⋯,zn=F0z1k,z2,⋯,zn⋅G¯z1k,z2,⋯,zn⋅diag0,1,⋯,1⋅V.
By Lemma3.2, we obtain that rank Fz1k,z2,⋯,zn is l−1. Hence, from (28), there is a normal full rank l−1×l submatrix Fkz2,⋯,zn≜Fkz1k,z2,⋯,zn of Fz1k,z2,⋯,zn such that
(29)Fkz2,⋯,zn=Az⋅diag0,1,⋯,1⋅V,where Az is an l−1×l submatrix of F0z1k,z2,⋯,zn⋅G¯z1k,z2,⋯,zn. By Lemma 3.5, the reduced minors of Fkz2,⋯,zn also generate the unit ideal Kz.
Sufficiency. If there exists a normal full rank l−1×l submatrix F1z2,⋯,zn≜F1z11,z2,⋯,zn of Fz11,z2,⋯,zn such that the reduced minors of F1z2,⋯,zn generate the unit ideal Kz, according to Lemma 3.1, we can construct a ZRP vector w1∈Kl×lz2,⋯,zn such that
(30)F1z2,⋯,zn⋅w1z2,⋯,zn=0,⋯,0T.
By Lemma3.2, Fz11,z2,⋯,zn is of normal rank l−1, from (30), and the fact that F1z2,⋯,zn is a normal full rank submatrix of Fz11,z2,⋯,zn, we have
(31)Fz11,z2,⋯,zn⋅w1z2,⋯,zn=0,⋯,0T.
By Quillen-Suslin Theorem [26], a unimodular square matrix V1∈Kl×lz2,⋯,zn can be constructed such that w1 is its first columns; combined with Lemma 2.1, we have
(32)Fz⋅V1z2,⋯,zn=A1z⋅diagz1−z11,1,⋯,1,where A1z∈Km×lz. It follows that
(33)Fz=A1z⋅G1z,where G1z=diagz1−z11,1,⋯,1⋅V1−1.It is obvious thatdetG1z=z1−z11, dA1=∏k=2pz1−z1k, and from (33), we have
(34)Fz12,z2,⋯,zn=A1z12,z2,⋯,zn⋅diagz12−z11,1,⋯,1⋅V1−1,(35)F2z12,z2,⋯,zn=A12z12,z2,⋯,zn⋅diagz12−z11,1,⋯,1⋅V1−1,where A12z12,z2,⋯,zn is an l−1×l submatrix of A1z12,z2,⋯,zn. By Lemma 3.2, rank Fz12,z2,⋯,zn is l−1 and from (34), rank A1z12,z2,⋯,zn is also l−1. By the assumption, the reduced minors of F2z12,z2,⋯,zn generate the unit ideal Kz. Since G1z12,1,⋯,1=diagz12−z11,1,⋯,1⋅V1−1 is unimodular, by (35) and Cauchy-Binet formula, we obtain that the reduced minors of A12z12,z2,⋯,zn also generate the unit ideal Kz.Repeating the procedure above forA1z with respect to z1−z12, we have
(36)A1z=A2z⋅G2zor
(37)Fz=A2z⋅G2z⋅G1z,such that A2z∈Km×lz, G2z=diagz1−z12,1,⋯,1⋅V2−1 with detG2z=z1−z12, V2 is a unimodular square matrix, dA2=∏k=3pz1−z1k. And
(38)F3z13,z2,⋯,zn=A23z13,z2,⋯,zn⋅diagz13−z12,1,⋯,1⋅V2−1⋅diagz13−z11,1,⋯,1⋅V1−1,where A23z13,z2,⋯,zn is an l−1×l submatrix of A2z13,z2,⋯,zn.Furthermore, sinceF3z13,z2,⋯,zn is of normal rank l−1 and the reduced minors of F3z13,z2,⋯,zn generate the unit ideal Kz, by (38), then A23z13,z2,⋯,zn is of normal rank l−1, and the reduced minors of A23z13,z2,⋯,zn also generate the unit ideal Kz.Repeating the same procedure continuously with respect toz1−z1k, k=3,…,p, we obtain that
(39)Fz=ApzGpz⋯G2zG1zor
(40)Fz=F0zG0z,where F0z=Apz∈Km×lz, G0z=∏k=1pGpz, detG0z=∏k=1pz1−z1k.Remark 3.3.
Similar to Remark3.2, judging whether Fz satisfies the conditions of Theorem 3.1 or not, we only need to compute the reduced Gröbner bases of some ideals.Example 3.1.
LetK≜Z/5 be the finite field, Fz1,z2,z3∈Kz1,z2,z3, and
(41)Fz1,z2,z3=2z14+4z13z22+2z13z3+4z1z3+3z22z3+4z22+4z2z33z2+4z33z12+z1z22+2z1z22+4z1z2+4z24+2z22z33z14z12z2+z1z22z3+3z1z32+2z1z3+4z22z3+z2z33z12+2z3.It is easy to check that the greatest common divisor of the2×2 minors of Fz is dF=z1+2z22+z3=z1−3z22+4z3. By computing, we have
(42)F3z22+4z3,z2,z3=4z22+4z2z3+z323z2+4z32z23+z22z3+z2z3+3z324z22+2z34z25+3z24z3+4z23z3+3z22z23+z2z32+z2z32z24+2z22z3+3z33+2z3.Obviously,F3z22+4z3,z2,z3 is of normal rank 1, the reduced minors of the first row of F3z22+4z3,z2,z3 (a 1×2 submatrix of F3z22+4z3,z2,z3) are 3z2+4z3 and 1 (since 4z22+4z2z3+z32=3z2+4z33z2+4z3), which generate the unit ideal Kz. By Theorem 3.1, there is a minor prime factorization for Fz1,z2,z3.To construct such a factorization, let(43)U=30z2+3z32,we have
(44)F3z22+4z3,z2,z3⋅U=0z2+3z303z22+4z304z24+4z22z3+z32+4z3,which gives
(45)Fz1,z2,z3⋅U=z13+2z3z2+3z34z1+z22z14z1z3+z3z12+4z3⋅z1+2z22+z3001.Let(46)F0z=z13+2z3z2+3z34z1+z22z14z1z3+z3z12+4z3,G0z=z1+2z22+z3001⋅204z2+2z33,we have
(47)Fz1,z2,z3=F0z1,z2,z3⋅G0z1,z2,z3.Remark 3.4.
In our two main theorems (Theorem3.1 and 3.3), Fz∈Km×ll≤m are of normal full rank. In fact, our methods can also deal with some matrices not of full normal rank. When the normal rank of Fz is r and r<l, if the reduce minors of Fz generate the unit ideal Kz, one has that Fz=Q1zB1z such that Q1∈Km×r of normal full rank, and B1∈Kr×l is ZRP. In order to consider the factorization of Fz, one needs to consider the factorization of Q1z. See the following example.Example 3.2.
LetK be the rational number field, Az∈K3×3z1,z2,z3, and
(48)Az=3z12−4z1z2−4z23z12z2−4z12z2+z1z2−1−3z12z2+4z1z2−z20z1−30z12z2z3−z1z2z3z13z2z3−z12z2z3+3z3−z12z2z3+2z1z3.By computing the minors ofAz, we have that rank Az=2, dA=z1−1z1−3, the reduced minors of Az generate the unit ideal Kz. Using Theorem 3.2 in [18], we have thatAz=FzBz, where
(49)Fz=4z1z2−4z2z1−13z1z2−9z2z1−3−z12z2z3+z1z2z33z3,Bz=−1z113z21+3z1z2z−3z2,and Bz is ZLP. It is easy to check that the greatest common divisor of the 2×2 minors of Fz is dF=z12−4z1+3=z1−1z1−3. Obviously, F1,z2,z3 is of normal rank 1, the reduced minors of the second row of F1,z2,z3 (a 1×2 submatrix of F1,z2,z3) are −6z2 and −2, which generate the unit ideal Kz. F3,z2,z3 is also of normal rank 1, the reduced minors of the first row of F3,z2,z3 are 8z2 and 8, which also generate the unit ideal Kz. By Theorem 3.3, there is a minor prime factorization for Fz1,z2,z3.To construct such a factorization, let(50)U1=10−3z21,we have
(51)F1,z2,z3=00−6z2−29z2z33z3,F1,z2,z3⋅U1=000−200,which gives
(52)Fz1,z2,z3⋅U1=−3z1z2+z2z12−10z1−3−z1z2z33z3⋅z1−1001.Thus,(53)Fz1,z2,z3=−3z1z2+z2z12−10z1−3−z1z2z33z3⋅z1−103z21.Let(54)F1z1,z2,z3=−3z1z2+z2z12−10z1−3−z1z2z33z3,U2=10z21,we have
(55)F13,z2,z3=−8z2800−3z2z33z3,F13,z2,z3⋅U2=080003z3,which gives
(56)F1z1,z2,z3⋅U2=z1z2z12−1z2z1−3−z2z33z3⋅z1−3001Thus,(57)F1z1,z2,z3=z1z2z12−1z2z1−3−z2z33z3⋅z1−30−z21,Fz1,z2,z3=z1z2z12−1z2z1−3−z2z33z3⋅z1−30−z21⋅z1−103z21.Let(58)F0z=z1z2z12−1z2z1−3−z2z33z3,G0z=z1−30−z21⋅z1−103z21,it is clear that detG0z=z12−4z1+3=dA. We have
(59)Fz1,z2,z3=F0z1,z2,z3⋅G0z1,z2,z3,Az1,z2,z3=F0z1,z2,z3⋅G0z1,z2,z3⋅Bz1,z2,z3,where Bz1,z2,z3 is ZLP.The desired minor prime factorization forAz1,z2,z3 is obtained.


## 4. Conclusions

In this paper, we have investigated the minor prime factorization for two classes ofn-D polynomial matrices whose g.c.d. of maximal order minors is z1−fz2,⋯,zn or ∏k=1pz1−z1k with their entries in Kz, where K is an arbitrary field. We have presented some key lemmas and two tractable criterions for the existence of minor prime factorizations. These criterions are easily checked by computing reduced Gröbner basis of the ideal generated by lower order minors, and the minor prime factorization can be carried out by Lin’s constructive methods.



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*Source: 6235649-2018-10-08.xml*