Minor Prime Factorization for <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2724395pt" id="M1" height="8.14084pt" version="1.1" viewBox="-0.0657574 -7.8684 8.77813 8.14084" width="8.77813pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"/></g></svg>-D Polynomial Matrices over Arbitrary Coefficient Field

Liu, Jinwang; Li, Dongmei; Zheng, Licui

doi:https://doi.org/10.1155/2018/6235649

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 6235649 | https://doi.org/10.1155/2018/6235649

Minor Prime Factorization for -D Polynomial Matrices over Arbitrary Coefficient Field

Jinwang Liu,¹Dongmei Li,¹and Licui Zheng¹

Academic Editor: Eulalia Martínez

Received11 Jan 2018

Revised04 Sept 2018

Accepted10 Sept 2018

Published08 Oct 2018

Abstract

In this paper, we investigate two classes of multivariate (-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.

1. Introduction

In the polynomial approach, pioneered by Rosenbrock [1], matrices over , 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 (-D) polynomial matrix has attracted much attention over the past decades because of its diverse applications in -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 -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 -D polynomial matrices whose greatest common divisor of maximal order minors is . 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 -D polynomial matrices whose coefficient field is arbitrary and minor prime factorizations for two classes of -D polynomial matrices whose the greatest common divisor of maximal order minors is or . 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 field , let denote the set of polynomial ring in variables with coefficients in and denote an algebraic closed field of . For the convenience, is denoted simply by ; when is the complex number field, is denoted simply by . Let denote the set of matrices with entries in and denote the zero matrices. Throughout the paper, the argument is omitted whenever its omission does not cause confusion.

Definition 2.1 (see [21]). Let be of normal rank and denote all the minors of the matrix . Extracting the greatest common divisor (g.c.d.), denoted by or , of gives Then, are called the reduced minors of .

Definition 2.2. Let be of normal full rank, then is said to be zero left prime (zero right prime) if the minors of generate the unit ideal , which is simplified as that is ZLP (ZRP).
First, the following two lemmas and a well-known result (Theorem 1 in [14]) are required.

Lemma 2.1. Let and . Supposing that , then is a divisor of .
The proof is simple and omitted.

Lemma 2.2. Let be polynomials, then they have no common zeros in (are zero coprime) if and only if there are such that i.e., is a ZLP row vector in , or generate the unit ideal .

Proof 1. Necessity. It is clear that . Let and denote the ideal generated by in and in , respectively. From the assumption and Hilbert Zero Theorem, . Then the unit element 1 (in ) may be represented by a Gröbner basis of , i.e., 1 may be reduced to 0 by . According to computational methods of , we have that , then 1 can be represented by in , i.e., there are such that Sufficiency. We assume that have common zeros in and is one of them. From (2), we have This is a contradiction. So have no common zeros in .

Theorem 2.1 (see [14]). Let , . Then a necessary and sufficient condition for to admit a minor prime factorization with , and is that there exists a full rank submatrix of such that the reduced minors of are zero coprime.

3. Main Results

For the complex number field and an -D polynomial matrix , from Lemma 2.2, we see that the reduced minors of are zero coprime if and only if the reduced minors of generate the unit ideal . In the following, we investigate the case in which , where 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. Let be of normal full rank, if the reduced minors of generate , then there is a ZLP matrix such that .

Proof 2. Since the reduced minors of generate the unit ideal , by Lin-Bose Theorem (Theorem 3.3 in [18]), can be factorized as such that is of normal full rank, and is ZRP. By Quillen-Suslin Theorem [26], there is a unimodular such that is its first rows, i.e., , where . Let , where , , then . Obviously, is ZLP. We have that .

Lemma 3.2. Let and , where for . Then rank is , where .

Proof 3. Let , for , we may write as where , .
Setting , , we have that . Since , then there is an minor of such that is a simple divisor of . Let denote an submatrix consisting of the rows of such that is a simple divisor of . Then Since , where , , is independent of , and is a simple divisor of , by the properties of determinant, some exists such that From (7) and Laplace Theorem, there is a permutation of such that It is obvious that , then is an minor of . Since , combined with (8), we have that rank is .

Lemma 3.3. Let and , where for . Then rank for every , and rank ; furthermore, the minors of generate the unit ideal , where .

Proof 4. For arbitrary but fixed , where , let , then are distinguished. Assuming that , for , we may write as where , .
Setting , , then we have that , Since , where , and , is a simple divisor of , by the properties of determinant, there is some such that From (11) and Laplace Theorem, there is a permutation of such that It is obvious that , then is an minor of . Since , combined with (12), we have that rank is for every , i.e., the minor of have no common zeros in . Noting that , then rank . Furthermore, by Lemma 2.2, we obtain that the minors of generate the unit ideal .

Corollary 3.1. Let with , where are distinguished. Then rank , the minors of generate the unit ideal , where .

Lemma 3.4. Let , , where for . Then, for , where , is an invertible matrix.

Proof 5. Since , for , by Lemma 3.3, rank and the minors of generate . By Lemma 3.1, there is a ZLP vector such that . According to Quillen-Suslin Theorem [26], a unimodular matrix can be constructed such that is its first column, then where . Setting , then the desired results are obtained.

Lemma 3.5. Let , if is of normal full rank and has the following factorization: where , is a unimodular matrix. Then the reduced minors of generate the unit ideal .

Proof 6. We partition , as the following form, respectively, where , , , . Then is ZLP, i.e., the reduced minors of generate the unit ideal , and So the reduced minors of are the same as that of , which generate the unit ideal .

Theorem 3.1. Let be an arbitrary field, be of normal full rank , . Then a necessary and sufficient condition for to admit a minor prime factorization with , and is that there exists a normal full rank submatrix of such that the reduced minors of generate the unit ideal .

Proof 7. Necessity. Supposing that admits a minor prime factorization with , , and . From Lemma 3.4, , where is unimodular. Combined with (18), we have By Lemma 3.2, we see that the normal rank of is . Hence, from (19), there exists a normal full rank submatrix of such that where is a submatrix of . By Lemma 3.5, the reduced minors of generate the unit ideal .
Sufficieny. Assuming that there exists a normal full rank submatrix of such that the reduced minors of generate the unit ideal . According to Lemma 3.1, a ZRP vector can be constructed such that Since is of normal rank , by (21) and the fact that is a normal full rank submatrix of , we have By Quillen-Suslin Theorem [26], a unimodular square matrix can be constructed such that is its first column; combined with Lemma 2.1, we have where . It follows that where with .

Remark 3.1. When 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 Theorem 3.1, in order to check that satisfies the conditions of Theorem 3.1, we first find all the submatrices of , denoted by , where . For every , we compute the minors of , saying as . Let the ideal , we compute the reduced Gröbner basis of . If contains only a polynomial , by the following result (Theorem 3.2), the reduced minors of generate the unit ideal .

Theorem 3.2. Let be the greatest common divisor of , i.e., and be the reduced Gröbner basis of the ideal . Then contains only a polynomial if and only if generate the unit .

Proof 8. Necessity. Assuming that , it is obvious that , where . Then . Setting , since , we have that there are such that So . Thus, generate the unit ideal .
Sufficiency. Assuming that generate the unit ideal , then the reduced Gröbner basis of the ideal is . Furthermore, the reduced Gröbner basis of is .
The reduced Gröbner basis algorithm of an ideal can be found in [16].

Remark 3.2. From the discussion above, we see that judging whether satisfies the conditions of Theorem 3.1 or not, we only need to compute the reduced Gröbner basis of , where .

Lin et al. studied the matrix factorization for a class of matrix whose g.c.d. of maximal order minors has divisor ([13]); in the following, we study the minor prime factorization for this class of matrices and propose a tractable criterion.

Theorem 3.3. Let be an arbitrary field, be of normal full rank, , where are distinguished. Then admits a minor prime factorization with , , and if and only if there are normal full rank submatrices of such that the reduced minors of generate the unit ideal , where .

Proof 9. Necessity. Supposing that admits a minor prime factorization as follows: where , , and . For arbitrary but fixed , from (27) and Lemma 3.4, we have By Lemma 3.2, we obtain that rank is . Hence, from (28), there is a normal full rank submatrix of such that where is an submatrix of . By Lemma 3.5, the reduced minors of also generate the unit ideal .
Sufficiency. If there exists a normal full rank submatrix of such that the reduced minors of generate the unit ideal , according to Lemma 3.1, we can construct a ZRP vector such that By Lemma 3.2, is of normal rank , from (30), and the fact that is a normal full rank submatrix of , we have By Quillen-Suslin Theorem [26], a unimodular square matrix can be constructed such that is its first columns; combined with Lemma 2.1, we have where . It follows that where .

It is obvious that , , and from (33), we have where is an submatrix of . By Lemma 3.2, rank is and from (34), rank is also . By the assumption, the reduced minors of generate the unit ideal . Since is unimodular, by (35) and Cauchy-Binet formula, we obtain that the reduced minors of also generate the unit ideal .

Repeating the procedure above for with respect to , we have or such that , with , is a unimodular square matrix, . And where is an submatrix of .

Furthermore, since is of normal rank and the reduced minors of generate the unit ideal , by (38), then is of normal rank , and the reduced minors of also generate the unit ideal .

Repeating the same procedure continuously with respect to , , we obtain that or where , , .

Remark 3.3. Similar to Remark 3.2, judging whether satisfies the conditions of Theorem 3.1 or not, we only need to compute the reduced Gröbner bases of some ideals.

Example 3.1. Let be the finite field, , and

It is easy to check that the greatest common divisor of the minors of is . By computing, we have

Obviously, is of normal rank 1, the reduced minors of the first row of (a submatrix of ) are and 1 (since ), which generate the unit ideal . By Theorem 3.1, there is a minor prime factorization for .

To construct such a factorization, let we have which gives

Let we have

Remark 3.4. In our two main theorems (Theorem 3.1 and 3.3), 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 is and , if the reduce minors of generate the unit ideal , one has that such that of normal full rank, and is ZRP. In order to consider the factorization of , one needs to consider the factorization of . See the following example.

Example 3.2. Let be the rational number field, , and

By computing the minors of , we have that rank , , the reduced minors of generate the unit ideal . Using Theorem 3.2 in [18], we have that, where and is ZLP. It is easy to check that the greatest common divisor of the minors of is . Obviously, is of normal rank 1, the reduced minors of the second row of (a submatrix of ) are and −2, which generate the unit ideal . is also of normal rank 1, the reduced minors of the first row of are and 8, which also generate the unit ideal . By Theorem 3.3, there is a minor prime factorization for .

To construct such a factorization, let we have which gives

Thus,

Let we have which gives

Thus,

Let it is clear that . We have where is ZLP.

The desired minor prime factorization for is obtained.

4. Conclusions

In this paper, we have investigated the minor prime factorization for two classes of -D polynomial matrices whose g.c.d. of maximal order minors is or with their entries in , where 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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors are grateful to the anonymous referees for instructive and helpful comments on this paper. This research is supported by the National Science Foundation of China (11871207, 11501192 and 11471108) and the Scientific Research Fund of Hunan Province Education Department (2017JJ3084).

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Copyright © 2018 Jinwang Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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