# On minor prime factorizations for multivariate polynomial matrices

- 75 Downloads

## Abstract

Multivariate polynomial matrix factorizations have been widely investigated during the past years due to the fundamental importance in the areas of multidimensional systems and signal processing. In this paper, minor prime factorizations for multivariate polynomial matrices are studied. We give a necessary and sufficient condition for the existence of a minor left prime factorization for a multivariate polynomial matrix. This result is a generalization of a theorem in Wang and Kwong (Math Control Signals Syst 17(4):297–311, 2005). On the basis of this result and a method in Fabiańska and Quadrat (Radon Ser Comp Appl Math 3:23–106, 2007), we give an algorithm to decide if a multivariate polynomial matrix has minor left prime factorizations and compute one if they exist.

## Keywords

Multidimensional systems Multivariate polynomial matrices Matrix factorizations Minor prime factorizations## Notes

### Acknowledgements

The authors would like to thank the reviewers whose valuable and constructive comments helped to improve this paper.

## References

- Adams, W. W., & Loustaunau, P. (1994).
*An introduction to Gröbner bases*. Providence: American Mathematical Society.CrossRefzbMATHGoogle Scholar - Bose, N. K., Buchberger, B., & Guiver, J. P. (2003).
*Applied multidimensional systems theory*. Dordrecht: Kluwer.Google Scholar - Brown, W. C. (1993).
*Matrices over commutative rings*. New York: Marcel Dekker Inc.zbMATHGoogle Scholar - Decker, W., Greuel, G. M., Pfister, G., & Schönemann, H. (2015). Singular 4-0-2—A computer algebra system for polynomial computations. http://www.singular.uni-kl.de. Accessed 20 Dec 2016.
- Eisenbud, D. (2013).
*Commutative algebra: with a view toward algebraic geometry*. New York: Springer.zbMATHGoogle Scholar - Fabiańska, A., & Quadrat, A. (2007). Applications of the Quillen-Suslin theorem to multidimensional systems theory.
*Radon Series Computational and Applied Mathematics*,*3*, 23–106.MathSciNetzbMATHGoogle Scholar - Fornasini, E., & Valcher, M. E. (1997). \(n\)-D polynomial matrices with applications to multidimensional signal analysis.
*Multidimensional Systems and Signal Process*,*29*, 387–408.MathSciNetCrossRefzbMATHGoogle Scholar - Guiver, J. P., & Bose, N. K. (1982). Polynomial matrix primitive factorization over arbitrary coefficient field and related results.
*IEEE Transactions on Circuits and Systems CAS*,*29*(10), 649–657.MathSciNetCrossRefzbMATHGoogle Scholar - Lin, Z. (1999). Notes on \(n\)-D polynomial matrix factorization.
*Multidimensional Systems and Signal Process.*,*10*(4), 379–393.MathSciNetCrossRefzbMATHGoogle Scholar - Lin, Z., & Bose, N. K. (2001). A generalization of Serre’s conjecture and some related issues.
*Linear Algebra and its Applications*,*338*, 125–138.MathSciNetCrossRefzbMATHGoogle Scholar - Lin, Z., Xu, L., & Fan, H. (2005). On minor prime factorization for \(n\)-D polynomial matrices.
*IEEE Transactions on Circuits and Systems II: Express Briefs*,*52*(9), 568–571.CrossRefGoogle Scholar - Liu, J., & Wang, M. (2015). Further remarks on multivariate polynomial matrix factorizations.
*Linear Algebra and its Applications*,*465*, 204–213.MathSciNetCrossRefzbMATHGoogle Scholar - Matsumura, H., & Reid, M. (1989).
*Commutative ring theory*. Cambridge: Cambridge University Press.Google Scholar - Morf, M., Levy, B. C., & Kung, S. Y. (1977). New results in 2-D systems theory, Part I: 2-D polynomial matrices, factorization, and coprimeness.
*Proceedings of the IEEE*,*65*(4), 861–872.CrossRefGoogle Scholar - Pommaret, J. F. (2001). Solving Bose conjecture on linear multidimensional systems. In
*Proceedings of the European control conference*(pp. 1853–1855).Google Scholar - Pommaret, J. F., & Quadrat, A. (1999). Algebraic analysis of linear multidimensional control systems.
*IMA Journal of Mathematical Control and Information*,*16*, 275–297.MathSciNetCrossRefzbMATHGoogle Scholar - Quadrat, A. (2003). The fractional representation approach to synthesis problems: An algebraic analysis viewpoint part I: (Weakly) doubly coprime factorizations.
*SIAM Journal on Control and Optimization*,*42*(1), 266–299.MathSciNetCrossRefzbMATHGoogle Scholar - Quadrat, A. (2013). Grade filtration of linear functional systems.
*Acta Applicandae Mathematicae*,*127*, 27–86.MathSciNetCrossRefzbMATHGoogle Scholar - Rotman, J. J. (2008).
*An introduction to homological algebra*. New York: Springer.zbMATHGoogle Scholar - Srinivas, V. (2004). A generalized Serre problem.
*Journal of Algebra*,*278*(2), 621–627.MathSciNetCrossRefzbMATHGoogle Scholar - Wang, M. (2007). On factor prime factorizations for \(n\)-D polynomial matrices.
*IEEE Transactions on Circuits and Systems I: Regular Papers*,*54*(6), 1398–1405.MathSciNetCrossRefzbMATHGoogle Scholar - Wang, M., & Feng, D. (2004). On Lin–Bose problem.
*Linear Algebra and its Applications*,*390*, 279–285.MathSciNetCrossRefzbMATHGoogle Scholar - Wang, M., & Kwong, C. P. (2005). On multivariate polynomial matrix factorization problems.
*Mathematics of Control, Signals, and Systems*,*17*(4), 297–311.MathSciNetCrossRefzbMATHGoogle Scholar - Youla, D. C., & Gnavi, G. (1979). Notes on \(n\)-dimensional system theory.
*IEEE Transactions on Circuits and Systems*,*26*, 105–111.MathSciNetCrossRefzbMATHGoogle Scholar