Skip to main content
SpringerLink
Log in

On Minor Left Prime Factorization Problem for Multivariate Polynomial Matrices

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

A new necessary and sufficient condition for the existence of minor left prime factorizations of multivariate polynomial matrices without full row rank is presented. The key idea is to establish a relationship between a matrix and any of its full row rank submatrices. Based on the new result, the authors propose an algorithm for factorizing matrices and have implemented it on the computer algebra system Maple. Two examples are given to illustrate the effectiveness of the algorithm, and experimental data shows that the algorithm is efficient.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bose N K, Applied Multidimensional Systems Theory, Van Nostrand Reinhold, New York, 1982.

    Google Scholar 

  2. Bose N K, Buchberger B, and Guiver J, Multidimensional Systems Theory and Applications, Dordrecht, The Netherlands: Kluwer, 2003.

    Google Scholar 

  3. Morf M, Levy B, and Kung S, New results in 2-D systems theory: Part I, 2-D polynomial matrices, factorization, and coprimeness, Proceeding of the IEEE, 1977, 65): 861–872.

    Article  Google Scholar 

  4. Guiver J and Bose N K, Polynomial matrix primitive factorization over arbitrary coefficient field and related results, IEEE Transactions on Circuits and Systems, 1982, 29(10): 649–657.

    Article  MathSciNet  Google Scholar 

  5. Liu J and Wang M, New results on multivariate polynomial matrix factorizations, Linear Algebra and Its Applications, 2013, 438(1): 87–95.

    Article  MathSciNet  Google Scholar 

  6. Youla D and Gnavi G, Notes on n-dimensional system theory, IEEE Transactions on Circuits and Systems, 1979, 26(2): 105–111.

    Article  MathSciNet  Google Scholar 

  7. Lin Z, Notes on n-D polynomial matrix factorizations, Multidimensional Systems and Signal Processing, 1999, 10(4): 379–393.

    Article  MathSciNet  Google Scholar 

  8. Charoenlarpnopparut C and Bose N K, Multidimensional FIR filter bank design using Gröbner bases, IEEE Transactions on Circuits and Systems II: Analog. Digit. Signal Processing, 1999, 46(12): 1475–1486.

    Google Scholar 

  9. Lin Z, Xu L, and Bose N K, A tutorial on Gröbner bases with applications in signals and systems, IEEE Transactions on Circuits and Systems I: Regular Papers, 2008, 55(1): 445–461.

    Article  MathSciNet  Google Scholar 

  10. Lin Z and Bose N K, A generalization of Serre’s conjecture and some related issues, Linear Algebra and Its Applications, 2001, 338(1): 125–138.

    Article  MathSciNet  Google Scholar 

  11. Pommaret J F, Solving Bose conjecture on linear multidimensional systems, Proceedings of European Control Conference, IEEE, Porto, Portugal, 2001, 1653–1655.

    Google Scholar 

  12. Wang M and Feng D, On Lin-Bose problem, Linear Algebra and Its Applications, 2004, 390(1): 279–285.

    Article  MathSciNet  Google Scholar 

  13. Wang M and Kwong C P, On multivariate polynomial matrix factorization problems, Mathematics of Control, Signals, and Systems, 2005, 17(4): 297–311.

    Article  MathSciNet  Google Scholar 

  14. Wang M, On factor prime factorization for n-D polynomial matrices, IEEE Transactions on Circuits and Systems, 2007, 54(6): 1398–1405.

    Article  MathSciNet  Google Scholar 

  15. Guan J, Li W, and Ouyang B, On rank factorizations and factor prime factorizations for multivariate polynomial matrices, Journal of Systems Science & Complexity, 2018, 31(6): 1647–1658.

    Article  MathSciNet  Google Scholar 

  16. Guan J, Li W, and Ouyang B, On minor prime factorizations for multivariate polynomial matrices, Multidimensional Systems and Signal Processing, 2019, 30): 493–502.

    Article  MathSciNet  Google Scholar 

  17. Lu D, Wang D K, and Xiao F H, On factor left prime factorization problems for multivariate polynomial matrices, Multidimensional Systems and Signal Processing, 2021, 32): 975–992.

    Article  MathSciNet  Google Scholar 

  18. Matsumura H and Reid M, Commutative Ring Theory, Cambridge University Press, Cambridge, 1989.

    Google Scholar 

  19. Mayr E W and Meyer A R, The complexity of the word problems for commutative semigroups and polynomial ideals, Advances in Mathematics, 1982, 46(3): 305–329.

    Article  MathSciNet  Google Scholar 

  20. Möller H M and Mora F, Upper and lower bounds for the degree of Gröbner bases, EUROSAM 1984, Ed. by Fith J, Lecture Notes in Computer Science, Springer-Verlag, New York, 1984, 174: 172–183.

    Google Scholar 

  21. Fabiańska A and Quadrat A, Applications of the Quillen-Suslin theorem to multidimensional systems theory, Eds. by Park H and Regensburger G, Gröbner Bases in Control Theory and Signal Processing, Radon Series on Computational and Applied Mathematics, 2007, 3): 23–106.

  22. Cox D, Little J, and O’shea D, Using Algebraic Geometry, Graduate Texts in Mathematics (Second Edition), Springer, New York, 2005.

    Google Scholar 

  23. Eisenbud D, Commutative Algebra: With a View Toward Algebraic Geometry, Springer, New York, 2013.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China

    Dong Lu

  2. KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Dingkang Wang

  3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Dingkang Wang

  4. MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, China

    Fanghui Xiao

Authors
  1. Dong Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dingkang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fanghui Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fanghui Xiao.

Ethics declarations

The authors declare no conflict of interest.

Additional information

This research was supported by the National Natural Science Foundation of China under Grant Nos. 12171469, 12001030 and 12201210, the National Key Research and Development Program under Grant No. 2020YFA0712300, and the Fundamental Research Funds for the Central Universities under Grant No. 2682022CX048.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lu, D., Wang, D. & Xiao, F. On Minor Left Prime Factorization Problem for Multivariate Polynomial Matrices. J Syst Sci Complex 37, 1295–1307 (2024). https://doi.org/10.1007/s11424-024-1367-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-024-1367-5

Keywords

Search

Navigation